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SAMPLE^  COPY. 
New  Rudiments  of  Arithmetic. 

PRICE. 

Fop  Introduction,  _______  ^O  cts 

Allowance  for  old  book  in  use  of  similar  grade,   "when 

given  in  exchange,      --__-_-      TO  ct». 


Books  ordered  for  introduction  will  be  delivered  at  above  named  prices 
in  any  part  of  the  United  States.  Sample  copies  for  examination,  with  a 
view  to  introduction,  will  be  sent  by  mail  to  teachers  or  school  ofjksrs  on 
receipt  of  the  introduction  price.        Address, 

Clark  &  Maynard, 

5  Barclay  St.,  New  York. 

(P.  O.  Box  1«19.) 


THOMSON'S    NEW     GRADED    SERIES. 


NEW 


RUDIMENTS 


OF 


AKITHMETIC 


COMBINING 


MENTAL  AND  SLATE  EXERCISES 


FOR 


INTERMEDIATE    DEPARTMENTS. 


By  JAMES  B.  THOMSON,  LL.  D., 

AUTHOR  OF  DAY  &  THOMSON'S  ARITHMETICAL  SERIES  ;    EDITOR  OF    DAY'i 
SCHOOL  ALGEBRA,  LEGENDRE'S  GEOMETRY,  ETC. 


NEW    YORK: 

CLAEK  &  MAYNARD,  PUBLISHERS, 

5  barclay  street. 

Chicago:    46  Madison  Street. 


• wn<oa, 

Thomson's"  Mathematical  Series. 
— ~~  ^ft. 

I.  A  Graded  Series  of  Arithmetics,  in  three  Books,  viz. : 

New  Illustrated  Table  Book,  or  Juvenile  Arithmetic.    With 
oral  and  slate  exercises.     (For  beginners.)     128  pp. 

New  Rudiments  of  Arithmetic.    Combining  Mental  with  Writ- 
ten Arithmetic.    (For  Intermediate  Classes.)    224  pp. 

New  Practical   Arithmetic.     Adapted  to  a  complete  business 
education.    (For  Grammar  Departments.)    384  pp. 

II.  Independent  Books. 

Key  to  New  Practical  Arithmetic.    Containing  many  valuable 
suggestions.     (For  teachers  only.)    168  pp. 

New  Mental  Arithmetic.      Containing  the  Simple  and  Com- 
pound Tables.     (For  Primary  Schools.)    144  pp. 

Complete  Intellectual  Arithmetic.    Specially  adapted  to  Classes 
in  Grammar  Schools  and  Academies.     168  pp. 

III.  Supplementary  Course. 

New  Practical  Algebra.     Adapted  to  High  Schools  and  Acad- 
emies.    312  pp. 

Key  to  New  Practical  Algebra.      With  full  solutions.      (For 
teachers  only.)    224  pp. 

New  Collegiate  Algebra.     (In  preparation.) 

Complete  Higher  Arithmetic.     (In  preparation.) 

##*  Each  book  of  the  Series  is  complete  in  itself. 

Copyright,  1872,  by  James  B.  Thomson. 

EDUCATION  DEPT. 


PREFACE. 


The  "New  Graded  Series,"  of  which  this  is  the  second 
book,  is  divided  into  three  parts.  The  object  of  this 
arrangement  is  convenience  and  economy. 

While  there  may  be  objections  to  an  "  indeterminate 
series  of  school-books,"  it  must  be  admitted  that  exer- 
cises in  reading,  arithmetic,  etc.,  which  are  adapted  to 
the  capacity  of  beginners,  are  totally  unfit  for  advanced 
classes.  In  view  of  this  fact,  it  requires  no  arguments  to 
show  that  a  "limited  series,"  adapted  to  the  different 
capacities  of  learners,  is  a  dictate  of  common  sense. 

Each  book  in  this  Series  is  complete  in  itself.  The  defini- 
tions and  principles,  so  far  as  each  extends,  are  expressed 
in  the  same  language,  but  the  examples  are  all  different. 

The  present  work  consists  of  a  course  of  Mental  and 
Written  Exercises  combined.    It  is  designed : 

ist.  To  develop  the  elementary  principles  of  the  science 
by  oral  examples. 

2d.  To  familiarize  the  pupil  with  the  application  of 
these  principles  to  the  solution  of  problems  requiring 
the  use  of  the  slate. 

3d.  To  lead  him  to  generalize  the  principles  thus 
developed,  and  to  put  the  steps  of  particular  solutions 
into  a  concise  statement,  or  General  Rule. 

4th.  To  secure  accuracy  and  rapidity  in  the  combina- 
tion of  numbers. 

Finally,  the  work  is  specially  adapted  to  intermediate 
classes,  who  are  beginning  to  "  cipher."  The  New  Rudi- 
ments, it  is  hoped,  may  facilitate  the  progress  of  pupils, 
and  merit  the  approval  of  teachers. 

James  B.  Thomson. 
New  York,  July.  1872. 


SUGGESTIONS. 


i.  Pakticular  attention  should  be  paid  to  the  assignment  of 
Lessons.  They  should  be  neither  too  long,  nor  too  short ;  but 
adapted  to  the  capacity  of  the  class,  and  the  time  they  have  for 
preparation. 

2.  Thoroughness  should  be  insisted  on,  at  every  step.  The 
acceptance  of  an  imperfect  lesson,  whether  from  sympathy,  or 
inattention,  is  a  positive  injury  to  the  pupil. 

3.  The  most  effective  auxiliaries  of  thoroughness  are  frequent 
reviews  and  Tabular  drills.    "  Practice  makes  perfect." 

4.  A  perfect  recitation  implies  both  promptitude  and  correctness. 
In  reciting  problems,  the  analysis  should  be  logical,  and  the  lan- 
guage correct. 

5.  Pupils  should  be  encouraged  to  study  out  different  solutions 
of  the  same  problem,  and  to  exercise  their  judgment  in  selecting 
the  most  simple,  logical,  and  concise. 

6.  Care  should  also  be  taken  to  prevent  the  Jidbit  of  adding  by 
counting  the  fingers.  Counting  is  not  addition.  Pupils  should 
be  taught  to  add  numbers  as  a  whole,  and  be  able  to  name  the 
sum  of  any  two  given  digits,  instantly. 

7.  The  definitions  should  be  carefully  explained,  and  thoroughly 
committed  to  memory.  Each  principle  and  rule  should  be  dwelt 
upon  until  the  pupil  comprehends  it,  and  is  able  to  give  a  correct 
account  of  it,  in  his  own  language,  or  that  of  the  author. 

8.  Cultivate  the  habit  of  self-reliance  in  the  solution  of  problems. 
It  is  better  for  the  pupil  to  solve  one  example,  independent  of  the 
answer  and  all  extraneous  aid,  than  a  dozen  by  the  help  of  a 
teacher,  or  a  key. 

9.  Special  pains  should  also  be  taken  to  cultivate  the  perceptive 
faculties,  and  correct  the  erroneous  ideas  of  learners  as  to  distance, 
surface,  weight,  etc. 

10.  In  developing  the  idea  of  Fractions,  and  the  Units  of  Weights 
and  Measures,  let  the  pupil  divide  some  object  into  halves,  thirds, 
etc.,  and,  if  possible,  let  him  see  and  handle  the  actual  stiutdardx 
of  length,  surface,  capacity,  and  weight.  These  simple  acts  will 
give  him  a  more  exact  idea  of  Fractions,  and  of  Weights  and 
Measures,  than  a  score  of  pictures,  or  a  talk  an  hour  long. 


CONTENTS. 


PAGE 

Number,         ..  ....--7 

dotation,  -------  8 

Arabic  Notation,         ------        8 

Roman  Notation,  -            -            -            -           -  '  13 

Numeration,           -           -           -          -          -  14 

Addition,            -           -           -           -           -           -  17 

When  the  sum  of  a  column  is  less  than  10,             -  -      24 

What  numbers  can  be  added  together,  -            -            -  25 

When  the  sum  of  a  column  is  10  or  more,  -            -  26 

Carrying  illustrated,         -            -            -            -            -  27 

General  Rule  for  Addition,  -            -            -            -  23 

Subtraction,       -          -           -           -           -           -  32 

When  each  figure  of  the  subtrahend  is  less  than  the  one 

above  it,           ------  37 

What  numbers  can  be  subtracted  one  from  another,  -      38 
When  a  figure  in  the  lower  number  is  larger  than  the 

one  above  it,                      -            -           -           -  41 

General  Rule  for  Subtraction,      -           -           -           -  42 

Drill  in  Rapid  Combinations,            -            -            -  44 

Multiplication,  -           -           -           -           -           -  45 

When  the  Multiplier  has  but  one  figure,     -            -  49 

When  the  Multiplier  has  two  or  more  figures,  -            -  53 

General  Rule  for  Multiplication,       -           -            -  -       54 

Contractions,         -           -           -            -            -            *  5° 

Questions  for  Review,             -           -           -            -  59 

Division,  -------  60 

Objects  of  Division  illustrated,          -            -            -  63 

Short  Division,      ------  64 

Long  Division,            -     .      -            -            -           -  70 

Contractions,          ------  73 

Drill  in  Rapid  Combinations,            -  "75 

Questions  for  Review,      .....  76 

Factoring,     •          -           -          •                     -  -79 


CONTENTS. 

PAGH 

Cancellation,     - 

81 

Greatest  Common  Divisors,  - 

-       83 

Least  Common  Multiple,  - 

86 

Fractions, 

-       88 

Notation  of  Fractions, 

90 

General  Principles  of  Fractions, 

-       93 

Reduction  of  Fractions,    - 

93 

Common  Denominator, 

-     101 

Least  Common  Denominator 

102 

Addition  of  Fractions, 

-     104 

Subtraction  of  Fractions,  - 

107 

Multiplication  of  Fractions,  - 

-    no 

Division  of  Fractions, 

116 

Reducing  Complex  Fractions  to 

Simple  ones,         -            -     122 

Questions  for  Review, 

124 

Fractional  Relation  of  Numbers, 

-     126 

Decimal  Fractions,    - 

129 

Notation  of  Decimals, 

-     130 

Reduction  of  Decimals,     - 

133 

Addition  of  Decimals, 

-     135 

Subtraction  of  Decimals,  - 

136 

Multiplication  of  Decimals,   - 

-     138 

Division  of  Decimals, 

140 

United  States  Money,    - 

-     143 

Addition  of  U.  S.  Money, 

147 

Subtraction  of  U.  S.  Money,  - 

-     149 

Multiplication  of  U.  S.  Money, 

150 

Division  of  U.  S.  Money, 

-     152 

Applications  of  U.  S.  Money, 

155 

Compound  Tables, 

-     158 

Reduction, 

i73 

Measurement  of  Rectangular  Su 

rfaces,        •           -           -    179 

Measurement  of  Rectangular  So^ 

ids,      -           -           -          180 

Denominate  Fractions, 

-    1S2 

Compound  Rules, 

-    185-191 

Percentage,  - 

-    193 

Profit  and  Loss,  - 

201 

Interest, 

-     204 

RUDIMENTS. 


DEFINITIONS 


1.  What  is  Arithmetic? 
Arithmetic  is  the  science  of  numbers. 

2.  What  is  a  single  thing  called  ? 

A  Unit,  or  One. 

3.  If  another  is  put  with  it,  what  ? 

Two. 

4.  If  another,  and  another,  etc.,  what  ? 

Three,  four,  five,  six,  etc. 

5.  What  is  number  ? 

Number  is  a  unit,  or  a  collection  of  units. 

6.  When  a  number  is  not  applied  to  any  object,  what  is  it 
called? 

An  Abstract  Number. 

7.  When  it  is  applied  to  some  object,  what  ? 

A  Concrete  Number.    Give  examples. 

8.  When   numbers  express  units  of   the  same  kind,  as,  3 
apples  and  4  apples,  5  and  7,  etc.,  what  are  they  called  ? 

Like  Numbers. 

9.  When  they  express  units  of  different  kinds,  as,  4  books 
and  6  pencils,  what  ? 

Unlike  Numbers. 


NOTATION 


10.  What  is  Notation  ? 

Notation  is  the  art  of  expressing  numbers  by 
figures,  letters,  or  other  numeral  characters. 

11.  What  are  the  two  principal  methods  in  use? 
The  Arabic  and  the  Roman. 

ARABIC    NOTATION. 

1.  What  i9  the  Arabic  Notation,  and  why  so  called  ? 

The  Arabic  Notation  is  the  method  of  express- 
ing numbers  by  figures  ;  and  is  so  called  because  it  was 
introduced  into  Europe  from  Arabia. 

2.  How  many  figures  does  it  employ,  and  what  called  ? 

Ten,  called — 

i,   2,    3,    4,    5,   6,    7,    -8,     9,     o. 

One,  two,  three,  four,  five,  six,  seven,  eight,  nine,  naught. 

3.  What  are  the  first  nine  called,  and  why  ? 

The  first  nine  are  called  significant  figures,  because 
each  of  them  always  expresses  a  number. 

They  are  also  called  digits,  from  digitus,  a  finger, 
because  the  ancients  used  to  reckon  upon  their  fingers. 

4.  What  is  the  last  called,  and  why  ? 

The  last  is  called  naught,  because  when  standing 
alone  it  has  no  value,  and  when  connected  with  sig- 
nificant figures,  it  denotes  the  absence  of  the  orders  iu 
whose  place  it  stands. 

It  is  also  called  zero,  or  cipher. 

Note. — The  pupil  should  learn  to  distinguish  and  write  tho 
Arabic  figures  with  readiness  before  proceeding  further. 

5.  How  is  each  of  the  first  nine  numbers  expressed  ? 

By  a  single  figure. 

jB.  What  are  these  numbers  called  ? 

Units  of  the  first  order,  or  simply  units. 


NOTATION.  9 

7.  What  is  the  greatest  number  expressed  by  one  figure? 

Nine* 

8.  How  is  ten  expressed? 

Ten  is  expressed  by  writing  i  in  the  second  place,  with 
a  cipher  on  the  right ;  as,  io. 

9.  What  is  the  I  called,  standing  in  the  second  place  ? 

A  unit  of  the  second  order. 

i.  Explain  and  write  each  of  the  numbers  from  ten 
to  twenty. 

Eleven  is  composed  of  one  ten  and  one  unit,  and  is 
expressed  by  writing  i  in  the  second  place  to  denote 
the  ten,  and  i  in  the  first  or  right  hand  place  to  denote 
the  unit ;  as,  n. 

Twelve  is  composed  of  one  ten  and  two  units,  and  is 
expressed  by  writing  i  in  the  second  place,  and  2  in 
the  first ;  as,  12. 

Tliirteen  is  composed  of  one  ten  and  three  units, 
and  is  expressed  by  writing  1  in  the  second  place,  and 
3  in  the  first;  as,  13,  etc. 

Twenty  is  two  tens,  and  is  expressed  by  placing  2 
in  the  second  place,  and  o  in  the  first ;  as,  20. 

2.  Explain  in  like  manner,  and  write  each  of  the 
numbers  from  twenty  to  thirty. 

3.  From  thirty  to  forty.    From  forty  to  fifty. 

4.  From  fifty  to  sixty.  From  sixty  to  seventy ;  and 
so  on  to  one  hundred. 

10.  What  is  the  greatest  number  that  can  be  expressed  by 
tioo  figures? 

Ninety-nine. 

11.  How  is  a  hundred  expressed? 

A  hundred  is  expressed  by  writing  1  in  the  third 
place,  with  tivo  ciphers  on  the  right ;  as,  100. 

12.  What  is  the  1  called,  standing  in  the  third  place? 

A  unit  of  the  third  order. 


10  NOTATION. 

5.  Explain  and  write  each  of  the  numbers  from  one 
hundred  to  one  hundred  and  ten. 

One  hundred  and  one  equals  one  hundred,  no  tens, 
and  one  unit,  and  is  expressed  by  writing  1  in  the  third 
place,  o  in  the  second,  and  1  in  the  first ;  as,  101. 

One  hundred  and  two  is  expressed  by  102 ;  one  hun 
dred  and  three  by  103,  etc.  , 

6.  Explain  and  write  each  number  from  one  hundred 
and  ten  to  one  hundred  and  twenty. 

One  hundred  and  ten  is  composed  of  one  hundred, 
one  ten,  and  no  units,  and  is  expressed  by  writing  1  in 
the  third  place,  1  in  the  second,  and  o  in  the  first;  as,  no. 

One  hundred  and  eleven  by  in;  one  hundred  and 
twelve  by  112,  etc. 

7.  Explain  in  like  manner,  and  write  the  numbers 
from  one  hundred  and  twenty  to  one  hundred  and 
thirty. 

8.  Write  the  numbers  from  one  hundred  and  thirty 
to  one  hundred  and  fifty.  From  one  hundred  and 
fifty  to  two  hundred. 

9.  Write  two  hundred.  Three  hundred.  Four  hun- 
dred.   Five  hundred. 

Write  the  following  numbers  in  figures : 

10.  One  hundred  and  twenty-three. 

1 1.  Two  hundred  and  thirty-seven. 

1 2.  Three  hundred  and  forty-five. 

13.  Four  hundred  and  ten. 

14.  Six  hundred  and  seven. 

15.  Five  hundred  and  sixty-three. 

16.  Six  hundred  and  five. 

1 7.  Seven  hundred  and  thirty. 

18.  Six  hundred  and  seventy-five. 

19.  Eight  hundred  and  forty-three. 

20.  Nine  hundred  and  ninety-nine. 


NOTATION.  11 

13.  What  is  the  largest  number  that  can  be  expressed  by 
three  figures  ? 

Nine  hundred  and  ninety-nine. 

Note. — The  preceding  exercises  should  be  repeated  and  sup- 
plemented by  dictation,  until  the  class  become  perfectly  familiar 
with  writing  numbers  less  than  a  thousand. 

1  -4.  How  are  numbers  larger  than  999  expressed  ? 

By  Other  Orders,  called  thousands,  tens  of  thou* 
Bands,  hundreds  of  thousands,  millions,  tens  of  millions, 
ate,  each  succeeding  order  having  ten  times  the  value 
of  the  preceding. 

1 5.  What  is  the  general  law  by  which  the  orders  of  units 
Increase  ? 

TJiey  increase  from  right  to  left  by  the  scale  of  ten ; 
that  is, 

Ten  simple  units  make  one  ten  ; 

Ten  tens  make  one  hundred  ; 

Ten  hundreds  make  one  thousand ;  and,  universally, 
ten  of  any  loiver  order  make  one  of  the  next  higher. 

16.  What  places  do  the  different  orders  occupy? 
Simple  units  occupy  the  right  hand  place ; 
Tens,  the  second  place ; 

Hundreds,  the  third  place ; 
Thousands,  the  fourth  place ; 
Tens  of  thousands,  the  fifth  place ; 
Hundreds  of  thousands,  the  sixth  place ; 
Millions,  the  seventh  place,  etc. ;  the  order  of  units 
corresponding  with  the  place  which  the  figure  occupies. 

17.  What  is  the  effect  of  moving  a  figure  from  right  to  left,  or 
from  left  to  right. 

Its  value  is  increased  tenfold  for  every  place  it  is 
moved  from  right  to  left ;  and  is  diminished  tenfold  for 
every  place  it  is  moved  from  left  to  right. 

18.  What  are  the  different  values  of  a  figure  called  ? 
The  simple  and  local  values. 


12  NOTATION-. 

1 9.  What  is  the  simple  value  of  a  figure  ? 

The  simple  value  of  a  figure  is  the  number  of 
units  it  expresses  when  it  stands  alone. 

20.  The  local  value  of  a  figure  ?    Illustrate  both. 

The  local  value  is  the  number  it  expresses  when 
connected  with  other  figures,  and  is  determined  by  the 
place  it  occupies,  counting  from  the  right 

21.  What  is  the  rule  for  expressing  numbers  by  figures  ? 
Begin  at  the  left  hand,  and  write  the  figures  of  the 

given  orders  in  their  successive  places  toicard  the  right. 

If  any  intermediate  orders  are  omitted,  supply  their 
places  ivith  ciphers. 

Write  the  following  numbers  in  figures : 

20.  One  thousand,  three  hundred,  and  sixty. 

2i.  Five  thousand,  seven  hundred,  and  thirty-five. 

22.  Seven  thousand,  three  hundred,  and  sixty-two. 

23.  Twenty-six  thousand  and  seventy-five. 

24.  Thirty-seven  thousand,  one  hundred,  and  six. 

25.  Ninety-five  thousand  and  seventeen. 

26.  One  hundred  and  twenty-three  thousand  and  two 
hundred. 

27.  Three  hundred  and  forty-eight  thousand  and  two 
hundred. 

28.  Four  hundred  and  ten  thousand,  three  hundred, 
and  forty. 

29.  Five  hundred  and  forty  thousand,  six  hundred, 
and  thirty. 

30.  Six  hundred  thousand,  two  hundred  and  forty. 

31.  Seven  hundred  and  fifty-five  thousand,  two  hun- 
dred, and  three. 

32.  Eight  hundred  and  fifty  thousand  and  three 
hundred. 

33.  Nine  hundred  and  thirty-eight  thousand  and 
sixty-eight. 


NOTATION. 


13 


ROMAN     NOTATION. 

22.  What  is  the  Roman  Notation  ;  and  why  so  called? 

The  Roman  Notation  is  the  method  of  express- 
ing numbers  by  letters ;  and  is  so  called  because  it  was 
employed  by  the  Romans. 

23.  How  many,  and  what  letters  are  used  ? 
Seven  capitals,  viz, :  I,  V,  X,  L,  C,  D,  and  M. 

24.  What  does  each  of  these  letters  express  ?  * 
The  letter  I  expresses  one ;  V,  five ;  X,  ten;  L,  fifty, 

C,  one  hundred ;  D,  five  hundred ;  and.  M,  one  thou* 
sand. 


TABLE. 

I 

denotes  one. 

XXIV 

denotes  twenty-four. 

II 

m 

two. 

XXV 

"      twenty-five. 

III 

" 

three. 

XXVI 

"      twenty-six. 

IV 

u 

four. 

XXVII 

"      twenty-seven. 

V 

a 

five. 

XXVIII 

"      twenty-eight. 

VI 

« 

six. 

XXIX 

twenty-nine. 

VII 

** 

seven. 

XXX 

"       thirty. 

VIII 

M 

eight. 

XL 

forty. 

IX 

« 

nine. 

L 

«       fifty. 

X 

M 

ten. 

LX 

"      sixty. 

XI 

M 

eleven. 

LXX 

"      seventy. 

XII 

« 

twelve. 

LXXX 

"      eighty. 

XIII 

t( 

thirteen. 

XC 

"      ninety. 

XIV 

« 

fourteen. 

C 

"      one  hundred. 

XV 

« 

fifteen. 

CC 

f      two  hundred. 

XVI 

<i 

sixteen. 

CCC 

"      three  hundred. 

XVII 

" 

seventeen. 

cccc 

M      four  hundred. 

XVIII 

U 

eighteen. 

D 

"      five  hundred. 

XIX 

t( 

nineteen. 

DC 

"      six  hundred. 

XX 

« 

twenty. 

DCC 

"      seven  hundred. 

XXI 

m 

twenty-one. 

DCCC, 

"      eight  hundred. 

XXII 

u 

twenty-two. 

DCCCC 

"      nine  hundred. 

XXIII 

11 

twenty-three. 

M 

"      one  thousand. 

MDCCCLXXII,  one  thousand, 

eight  hundred,  and  seventy-two. 

14 


NUMERATION. 


Notes.— i.  Repeating  a  letter  repeats  its  value.  Thus,  1  de- 
notes one  ;  II,  two ;  III,  three;  X,  ten  ;  XX,  twenty,  etc. 

2.  Placing  a  letter  of  less  value  fo/<?re  one  of  greater  value, 
diminishes  the  value  of  the  greater  by  that  of  the  less;  placing 
the  less  after  the  greater  increases  the  value  of  the  greater  by 
that  of  the  less.  Thus,  I  denotes  one,  and  V  five ;  but  IV  is  four 
and  VI  six. 

3.  Placing  a  Iwrizontal  line  over  a  letter  increases  its  value  a 
thousand  times.  Thus,  I  denotes  a  thousand ;  X,  ten  thousand ; 
C,  a  hundred  thousand  ;  M,  a  million. 

4.  Four  was  formerly  denoted  by  IIII ;  nine,  by  Villi ;  forty, 
by  XXXX ;  and  ninety,  by  LXXXX. 

Express  the  following  numbers  by  letters : 


I.  17. 

7. 48. 

13.  98. 

19.  564. 

2.  26. 

8.  53- 

14.  in. 

20.  896. 

3.  24. 

9.  67. 

15.  109. 

21.  1116. 

4-  37- 

10.  78. 

16.  114. 

22.  1320. 

5.  29. 

11.  84. 

17.  118. 

23-   1536. 

6.  44. 

12.  89. 

18.  377- 

24.  1876. 

NUMERATION 


25.  What  is  Numeration  ? 

Numeration  is  the  art  of  reading  numbers  ex- 
pressed by  figures,  letters,  or  other  numeral  characters. 

26.  How  read  numbers  expressed  by  figures? 

Divide  them  into  periods  of  three  figures  each,  count- 
ing from  the  right. 

Beginning  at  the  left  hand,  read  the  periods  in  suc- 
cession, and  add  the  name  to  each,except  the  last. 

27.  Why  is  the  name  of  the  last  period  omitted  ? 
Because  the  right  hand  period  always  denotes  simple 

units,  therefore  its  name  need  not  be  mentioned. 


2*  U  M  E  R  A  T I  0  :N\ 


15 


Repeat  the  Numeration  Table,  beginning  with  units. 
NUMERATION     TABLE. 


5    7 


2    3: 


1  »  S 

3  a  S 

&  £  e 

8  7  4, 


I 


W    H    b 
267. 


Period  V. 
Trillions. 


Period  IV. 
Billions. 


Period  IH. 
Millions. 


Period  II. 
Thousands. 


Period  I. 
Units. 


The  periods  in  the  Table  are  thus  read  :  298  trillions, 
570  billions,  923  millions,  874  thousand,  two  hundred 
and  sixty-seven. 

Note. — This  method  of  reading  numbers  is  commonly  ascribed 
to  the  French,  and  is  thence  called  the  French  Numeration. 

Others  ascribe  it  to  the  Italians,  and  thence  call  it  the  Italia  . 
Method. 

Copy  and  read  the  following : 


I.  107. 

13- 

10315- 

25- 

342^201 

2.   260. 

14. 

12065. 

26. 

23C0400 

3-  3i5- 

*5- 

24308. 

27. 

5408025 

4.  809. 

16. 

13020. 

28. 

4532°265 

5.  1020. 

17- 

20460. 

29. 

63205308 

6.  2405. 

18. 

35°°7- 

30- 

265310275 

7.  3200. 

19. 

40800. 

3i- 

8045007 

8.  5007. 

20. 

85408. 

32- 

925604300. 

9.  6080. 

21. 

115326. 

33- 

4260345809 

10.  7650. 

22. 

208065. 

34- 

61204400000 

11.  8075. 

23. 

400304. 

35- 

160240030045 

12.  9364. 

24. 

803025. 

36. 

407008300416, 

L6 

NUMERATION. 

Copy  and  read  the  following: 

i.  III. 

II.  LIX. 

21.  CIV. 

2.  VI. 

12.  LVIII. 

22.  cvin. 

3-  iv. 

13.  LXIX. 

23.  CXII. 

4.  VIII. 

14.  LIV. 

24.  CIX. 

5-  VII. 

15.  LXXIII. 

25.  CXL. 

6.  IX. 

16.  XLIX. 

26.  CLXXIX. 

7.  XIII. 

17.  LXXXIV. 

27.  ccxc. 

8.  XXXII. 

18.  LXXXVIII. 

28.  DXIX. 

9.  XXVIII. 

19.  xov. 

29.  MDCCCXI. 

0.  XLII. 

20.  XCIX. 

30.  MDCCCLXXV. 

Express  by  figures,  and  read  the  following  numbers : 

1.  Two  hundred  and  five  thousand,  six  hundred,  and 
ninety-one. 

2.  Eight  hundred  and  forty  thousand,  five  hundred, 
and  nine. 

3.  Two  millions,  four  hundred  thousand,  and  seventy. 

4.  Forty-five  millions,  sixty  thousand,  two  hundred, 
and  sixty. 

5.  Three  hundred  and  ninety  millions,  four  thousand, 
a_  ^  seventy-two. 

C   Six  hundred  millions,  forty -eight  thousand,  and  ten. 

7.  Five  billions,  six  hundred  and  ten  millions,  and 
three  lundred. 

8.  One  hundred  and  forty  billions,  and  thirty-five  mil- 
lions. , 

9.  Forty-five  millions,  seven  hundred  and  sixty  thou- 
sand. 

10.  Three  hundred  and  twenty-nine  trillions,  six 
hundred  and  thirty-seven  billions,  three  hundred  and 
forty  millions,  four  hundred  and  nineteen  thousand, 
two  hundred  and  eighty-four. 

Note. — Dictation  exercises  in  reading  and  writing  numbers 
should  be  continued,  till  the  class  is  perfectly  familiar  with  both. 


ADDITION. 


MENTAL    EXERCISES. 

To  Teachebs.— The  object  of  this  Exercise  is  to  teach  beginners  tha 
process  of  adding  two  digits  together.  If  young,  let  them  illustrate  the 
examples  by  counters  or  unit  marks. 

i.  If  you  have  i  apple,  and  I  give  you  i  apple  more, 
how  many  apples  will  you  have  ? 

"  One  apple  and  i  apple  more  are  2  apples/, 

2.  If  you  have  2  cents,  and  you  find  1  more,  how 
many  cents  will  you  have  ? 

"  Two  cents  and  1  cent  more  are  3  cents." 

3.  How  many  are  3  marbles  and  2  marbles  ? 

4.  Show  this  by  your  fingers. 

5.  Sarah  has  3  red  roses,  and  3  white  ones :  how  many 
roses  has  she  of  both  kinds  ?     Show  it. 

6.  If  an  orange  costs  5  cents,  and  a  lemon  4  cents, 
how  much  will  both  cost  ?     Show  it. 

7.  William  earned  6  cents  in  the  morning,  and  4  in 
the  afternoon  :  how  much  did  he  earn  in  both  ? 

8.  Sanford  obtained  6  credit  marks,  and  his  sister  6 : 
how  many  did  both  obtain  ?     Show  it. 

9.  If  I  gather  6  quarts  of  cherries,  and  buy  8  quarts, 
how  many  quarts  shall  I  have  ?     Show  it. 

10.  Joseph  picked  5  quarts  of  blackberries,  and  his 
brother  7  quarts:  how  many  quarts  did  both  pick ? 

11.  If  I  pay  8  dollars  for  a  barrel  of  flour,  and  7  dol- 
iars  for  a  ton  of  coal,  what  shall  I  pay  for  both  ? 

12.  A  teacher  received  two  bouquets,  one  containing 
8  flowers  and  the  other  10':  how  many  flowers  were 
there  in  both  ? 

13.  George  picked  12  peaches  from  one  tree,  and  3 
from  another :  how  many  did  he  pick  from  both  ? 


18 


ADDITION. 


ADDITION    TABLE 


2  and 

3 

and 

4  and 

5 

and 

i  are  3 

1 

are  4 

1  are  5 

1 

are  6 

2  "   4 

2 

"   5 

2  "   6 

2 

?   7 

3  "   5 

3 

"   6 

3  "   7 

3 

"   8 

4  "   6 

4 

«   7 

4  "   8 

4 

"   9 

5  "   7 

5 

"   8 

5  "   9 

5 

"  10 

6  "   8 

6 

"   9 

6  "  10 

6 

"  11 

7  "   9 

7 

"  10 

7  "  11 

7 

"  12 

8  *     io 

8 

"  11 

8  "  12 

8 

u     I3 

9  "  ii 

9 

"  12 

9  "  J3 

9 

«  14 

IO   "   12 

10 

"  13 

10  "  14 

10 

"  15 

6  and 

7 

and 

8  and 

9 

and 

i  are  7 

1 

are  8 

1  are  9 

1 

are  10 

2  "   8 

2 

"   9 

2  "  10 

2 

"  11 

3  *   9 

3 

"  10 

3  "  11 

3 

«     12 

4  "  10 

4 

"  11 

4  "  12 

4 

"  13 

5  "  ll 

5 

u     12 

5  "  i3 

5 

"  14 

6   "12 

6 

"  13 

6  "  14 

6 

"  15 

7  "  13 

7 

"  14 

7  "  15 

7 

«  16 

8  "  14 

8 

"  i5 

8  "  16 

8 

"  i7 

9  "  15 

9 

"  16 

9  "  17 

9 

«  18 

10  u     16 

10 

«  I7 

10  "  18 

10 

"  19 

1.  Show  by  counters  or  unit  marks  how  many  2 
added  to  3  will  make. 

2.  Show  how  many  2  added  to  4  will  make. 

3.  Show  how  many  2  added  to  5  will  make. 

4.  Show  how  many  3  added  to  4  will  make. 

5.  Show  how  many  3  added  to  5  will  make. 

6.  Show  how  many  4  added  to  5  will  make,  etc. 

***  Particular  care  should  be  taken  to  see  that  beginners 
understand  how  the  Addition  Table  is  constructed;  that  they 
fully  comprehend  the  results  of  adding  two  digits  together,  be- 
fore they  are  requirod  to  commit  them  to  memory. 


ADDITION.  19 

DEFINITIONS. 

1.  What  is  Addition  ? 

Addition  is  uniting  two  or  more  numbers  in  one. 

2.  What  is  the  number  obtained  by  addition  called  ? 

The  Sum  or  Amount 
'      Note. — The  sum  or  amount  contains  as  many  units  or  ones  as 
all  the  numbers  added. 

3.  When  the  numbers  to  be  added  are  the  same  denomination, 
what  is  the  operation  called  ? 

Simple  Addition. 

4.  How  is  Addition  denoted  ? 

By  a  perpendicular  cross  called  plus  (  +  ), 
placed  between  the  numbers  to  be  added.  Thus,  5+3 
shows  that  5  and  3  are  to  be  added  together,  and  is 
read,  "  5  plus  3,"  "  5  and  3,"  or  "  5  added  to  3." 

Note. — The  term  plus  is  a  Latin  word,  signifying  more,  or 
added  to. 

5.  How  is  the  equality  between  two  numbers  or  sets  of  num 
bers  denoted  ? 

By  two  short  parallel  lines,  called  the  sign  of 
equality  (=).  The  expression  5+3  =  8,  shows  that  5 
increased  by  3  equals  8,  and  is  read,  "  5  plus  3  equal  8," 
or  the  sum  of  "  5  plus  3  equals  8." 

Read  the  following  expressions :  7  +  2  =  9;  6  +  4  = 
7+3;  17 +3  +  5  =  I2  +  7  +  6;  21+8  +  9  =  12  + 
20  +  6 ;  30  +  3  +  20  =  40  +  13. 

MENTAL    EXERCISES. 
To  Teachers.— The  Mental  and  Slate  Exercises  are  intended  to  be  com- 
bined in  each  recitation.    Hence,  no  more  examples  should  be  assigned  to  a 
lesson,  than  the  class  can  thoroughly  master. 

i.  Henry  gave  4  cents  for  an  orange,  3  cents  for  a 
pear,  and  2  cents  for  an  apple :  how  many  cents  did  he 
give  for  all  ? 

Analysis. — 4  cents  and  3  cents  are  7  cents,  and  2  are  9  cents. 
Therefore,  be  gave  9  cents  for  all. 


20  ADDITION. 

2.  John  paid  5  cents  for  a  writing-book,  and  2  cents 
for  a  pen  :  how  many  cents  did  he  pay  for  both  ?  5  and 
what  make  7  ? 

3.  If  an  orange  costs  4  cents,  a  pear  3  cents,  and  a 
peach  2  cents,  what  will  all  three  cost  ? 

4.  George  gave  4  apricots  to  one  of  his  sisters,  3  to 
another,  and  5  to  another:  how  many  did  he  give 
to  all? 

5.  If  you  pick  up  4  apples  under  one  tree,  3  under 
another,  and  5  under  another,  how  many  apples  will 
you  have  ? 

6.  A  man  paid  3  dollars  for  a  cane,  5  dollars  for  an 
umbrella,  and  4  dollars  for  a  hat :  how  much  did  he 
pay  for  all  ? 

7.  How  many  are  5  cents,  and  3  cents,  and  2  cents, 
and  1  cent? 

8.  Sarah  paid  9  dollars  for  a  hat,  and  4  dollars  for  a 
parasol :  what  did  she  pay  for  both  ? 

9.  Count  by  twos  rapidly  to  60.  Thus,  two,  four,  six, 
eight,  ten,  twelve,  etc. 

10.  Count  by  threes  in  like  manner  to  60. 

11.  Count  by  fours  to  100. 

SLATE     EXERCISES. 
Columns  of  Single  Figures. 

1.  Write  and  add  2,  4,  5,  3,  6,  and  1,  upward  and 
downward  four  times. 

2 
Explanation. — Write  the  numbers  in  a  column, 

and  draw  a  line  under  it. 

First.  Beginning  at  the  bottom,  add  upward  ;  as,       -> 

one,  seven,  ten,  fifteen,  nineteen,  twenty-one.  Set  21  ~ 
under  the  column. 

Second.  Begin  at  the  top,  and  add  downward  ;  as,  _J_ 

two,  six,  eleven,  etc.  2 1  A  ns. 


ADDITION.  21 

Copy  and  add  the  following  upward  and  down  ward 
orally  till  rapidity  is  attained : 

(»•)        (3-)       (4-)        (5-)        (6.)        (7-)       (8.)       (9.) 


2 

3 

4 

5 

2 

3 

4 

5 

4 

2 

3 

2 

I 

2 

i 

3 

3 

4 

5 

3 

5 

4 

3 

2 

i 

2 

i 

2 

2 

3 

4 

5 

2 

I 

3 

3 

I 

i 

2 

2 

2 

2 

2 

i 

3 

2 

5 

4 

I 

3 

4 

2 

2 

3 

3 

3 

2 

2 

3 

3 

4 

4 

3 

4 

3 

I 

2 

2 

i 

2 

4 

5 

4 

5 

5 

4 

_5 

3 

5 

4 

%*  Care  should  be  taken  to  write  figures  with  neatness  and 
symmetry,  and  set  them  in  perpendicular  columns. 

MENTAL      EXERCISES, 
i.  There  are  5  ducks  in  one  pond,  6  in  another,  and 
7  in  another :  how  many  are  there  in  all  the  ponds  ? 

2.  A  man  picked  8  quarts  of  blackberries,  and  his  son 
6  quarts :  how  many  quarts  did  both  pick  ? 

3.  William  has  9  chickens  in  one  coop,  and   7  in 
another :  how  many  chickens  has  he  in  both  ? 

4.  7  and  what  make  16  ?     9  from  16  leaves  what  ? 

5.  How  many  are  7,  and  6,  and  5,  and  4,  and  8  ? 

6.  If  there  are  9  pears  in  one  dish,  and  8,  in  another, 
Aow  many  are  there  in  both  dishes  ? 

7.  8  and  what  make  17  ?     9  from  17  leaves  what  ? 

8.  Henry  solved  6  examples,  George  9,  and  Samuel 
8 :  how  many  did  they  all  solve  ? 

9.  Lucy  gave  her  teacher  7  roses,  Julia  9,  and  Hattie 
10 :  how  many  roses  had  the  teacher? 

10.  Count  by  fives  to  100,  with  rapidity. 

11.  Count  by  sixes  to  100,  in  like  manner. 


22  ADDITION. 

SLATE     EXERCISES, 
Copy  and  add  the  following  as  before : 


!•) 

(2.) 

(3.) 

(4.) 

(50 

(6.) 

(7.) 

(8.) 

6 

5 

4 

7 

8 

7 

4 

9 

3 

4 

5 

4 

6 

5 

7 

5 

5 

2 

6 

7 

2 

8 

9 

4 

4 

5 

7 

3 

8 

3 

i 

9 

2 

3 

4 

5 

4 

4 

3 

6 

5 

4 

7 

7 

8 

8 

9 

9 

4 

6 

3 

4 

5 

6 

7 

8 

6 

4 

7 

8 

9 

5 

6 

4 

3 

7 

6 

9 

4 

8 

4 

3 

7 

5 

3 

5 

6 

7 

8 

9 

MENTAL    EXERCISES. 

i.  Henry  had  5  cents,  and  earned  6  more:  how  many 
cents  had  he  then  ?    5  and  what  make  1 1  ? 

2.  William  picked  7  quarts  of  cherries,  and  his  brother 
6  quarts  :  how  many  quarts  did  both  pick  ? 

3.  5  and  what  make  12  ?     6  and  what  make  13  ? 

4.  A  farmer  had  8  cows,  and  bought  6  more :  how 
many  had  he  then  ? 

5.  If  you  pay  7  dollars  for  a  velocipede,  and  8  dollars 
for  an  overcoat,  what  will  both  cost  you  ? 

6.  6  and  what  make  14?     7  and  what  make  15  ? 

7.  If  Helen  pays  8  dollars  for  a  music-box,  and  9  dol- 
lars for  a  fur  cape,  what  will  both  cost  her  ? 

8.  How  many  are  9  dollars,  and  6  dollars,  and  7  dol- 
lars? 

9.  How  many  are  15  rods,  6  rods,  and  8  rods  ? 

10.  Count  by  sevens  to  100,  with  rapidity. 

11.  Count  by  eights  to  100,  in  like  manner. 


ADDITION.  23 

SLATE     EXERCISES. 
Copy  and  cAd,  the  following  upward  and  downward 
as  before : 


(I.) 

(»■)■ 

(3-) 

(4.) 

(5-) 

(6.) 

(7-) 

(8.) 

4 

3 

2 

9 

5 

6 

8 

9 

3 

4 

3 

8 

3 

8 

8 

8 

6 

2 

5 

4 

8 

9 

5 

4 

8 

3 

7 

3 

4 

7 

6 

7 

I 

i 

6 

i 

2 

4 

7 

5 

2 

6 

i 

7 

3 

5 

3 

6 

7 

7 

3 

6 

9 

3 

5 

7 

3 

2 

4 

5 

6 

7 

9 

8 

5 

7 

6 

8 

7 

6 

5 

7 

8 

6 

7 

4 

6 

7 

8 

9 

Write  in  columns,  and  add  the  following: 

9.  3  pounds,  4  pounds,  5  pounds,  2  pounds,  7  pounds, 
and  1  pound. 

10.  6  yards,  5  yards,  3  yards,  4  yards,  2  yards,  8  yards, 
and  7  yards. 

MENTAL    EXERCISES. 

1.  How  many  are  3  and  10  ?  13  and  10  ?  23  and  10  ? 
33  and  10  ?  43  and  10  ?  53  and  10  ?  63  and  10  ?  73 
and  10  ?     83  and  10  ?    93  and  10  ? 

2.  7  and  10  ?  17  and  10  ?  27  and  10  ?  37  and  10  ? 
47  and  10?  57  and  10?  67  and  10?  77  and  10?  87 
and  10  ?    97  and  10? 

3.  3  and  4?  13  and  4?  23  and  4?  33  and  4?  53 
and  4  ?    43  and  4  ?     63  and  4  ?    83  and  4  ?     93  and  4  ? 

4.  5  and  3  ?  15  and  3  ?  25  and  3  ?  45  and  3  ?  35 
and  3  ?  55  and  3  ?  65  and  3  ?  85  and  3  ?  75  and  3  ? 
95  and  3  ? 


24  ADDITION. 

5.  17  and  5  t    37  and  5  ?     27  and  5  ?    57  and  5  ?    47 
and  5  ?    67  and  5  ?    87  and  5  ?     77  and  5  ?    97  and  5  ? 

6.  15  and  6  ?     35  and  6  ?     25  and  6  ?    45  and  6  ?    65 
and  6  ?    55  and  6  ?    75  and  6  ?    95  and  6  ?     85  and  6  ? 

7.  18  and  7  ?     28  ivnd  7  ?    38  and  7  ?    48  and  7  ?    58 
and  7?    68  and  7?     78  and  7?    88  and  7  ?    98  and  7  ? 

8.  16  and  8  ?     26  and  8  ?     36  and  8  ?    46  and  8  ?    56 
and  8?    66  and  8?    76  and  8?    86  and  8?    96  and  8? 

9.*  14  and  9  ?     24  and  9  ?    34  and  9  ?    44  and  9  ?    54 
and  9  ?    64  and  9  ?     74  and  9  ?    84  and  9  ?    94  and  9  ? 

10.  Count  by  nines  to  100,  with  rapidity. 

11.  Count  by  tens  to  100,  in  like  manner. 

SLATE     EXERCISES. 

When  the  Sum  of  a  Column  is  Less  than  10. 

1.  What  is  the  sum  of  234  dollars,  423  dollars,  and 
132  dollars  ? 

Analysis. — Wri  te  the  numbers  one  under  another,   Operation. 
the  units  figures  in  one  column,  the  tens  in  the  next,      234 
and  so  on.    Begin  at  the  right  and  add :    2  units      423 
and  3  units  are  5  units,  and  4  are  9  units.     Set  the      i<,2 

9    under  the    units    column,  because  it  is  units.      

Next,  3  tens  and  2  tens  are  5  tens,  and  3  are  8  tens.      7°9  tlols. 
Set  the  8  under  the  tens  column,  because  it  is  tens. 
Finally,  1  hundred  and  4  hundred  are  5  hundred,  and  2  are  7  hun- 
dred.    Set  the  7  under  the  hundreds,  because  it  is  hundreds. 

Note.— In  practice,  it  is  better  simply  to  pronounce  the  results ; 
as,  two,  five,  nine,  etc. 

Copy  and  add  the  following,  in  like  manner: 

(2.)   (3.)   (4.)   (5.)   (6.)   (70    (8.)  (90 

21    32    14   43    32    321    124  434 

34    11    23    2I    2°    i32    53i  243 

12    45    42    34    45    434    233  322 


ADDITION.  25 

Write  in  columns,  and  add  the  following : 

10.  25  +  12+30.  13.  34+21+40. 

11.  31  +  22  +  24.  14.46  +  30  +  12. 
12.40  +  13  +  36.                        15.  51  +  25  +  23. 

6.  How  do  you  write  numbers  to  be  added  ? 
Write  units  under  units,  tens  under  tens,  etc. 

7.  Where  begin  to  add,  and  how  proceed  ? 

Begin  at  the  right,  and  add  each  column  separately. 

8.  When  the  sum  of  a  column  is  less  than  10,  what  is  done 
with  it  ;  and  why  ? 

Set  it  under  the  column  added;  because  it  is  the 
same  order  as  that  column. 

9.  What  two  principles  are  necessary  to  be  observed  in 
addition  ? 

1  st.  The  numbers  must  be  Like  numbers. 
2d.  Units  of  the  same  order  must  be  added, 
each  to  each. 

Can  3  books  be  added  to  5  pencils  ?    Why  ? 

Do  4  units  and  3  tens  make  7  units,  or  7  tens  ?    Why  ? 

MENTAL    EXERCISES. 

1.  A  certain  school  had  40  girls  and  30  boys  in 
attendance :  how  many  pupils  did  it  contain  ? 

Analysis. — 40  is  equal  to  4  tens,  and  30  is  equal  to  3  tens. 
Now  4  tens  and  3  tens  are  7  tens,  or  70.  The  school,  therefore, 
contained  70  pupils. 

2.  50  is  how  many  tens?    40?    60?    70?     80?    100? 

3.  5  tens  are  how  many  ?  7  tens?  6  tens?  9  tens? 
8  tens?     10  tens? 

4.  How  many  are  3  tens  and  5  tens  ?  6  tens  and  4 
tens  ?    5  tens  and  7  tens  ? 

5.  How  many  are  20  and  30  ?     50  and  80  ? 

6.  How  many  are  70  and  50  ?     80  and  90  ? 

2 


26  ADDITION. 

7.  In  a  certain  grove  there  are  89  sugar-maples  and 
46  elms :  how  many  trees  are  there  in  the  groye  ? 

Analysis. — 89  is  8  tens  and  9  units,  and  46  is  4  tens  and  6 
units.  Now  8  tens  and  4  tens  are  12  tens,  or  120 ;  and  9  units 
and  6  units  are  15  units,  which,  added  to  120,  make  135.  There- 
fore, there  are  135  trees  in  the  grove. 

Note. — When  numbers  to  be  added  mentally  are  large,  it  is 
advisable  to  separate  them  into  the  units,  tens,  etc.,  of  which  they 
are  composed,  and  begin  to  add  with  the  highest  order. 

8.  How  many  are  34  and  43  ?     26  and  51  ? 

9.  How  many  are  45  and  62  ?     71  and  46  ? 

10.  A  man  paid  68  dollars  for  a  cow.  and  55  dollars 
for  a  colt :  what  did  he  pay  for  both  ? 


SLATE    EXERCISES. 
When  the  Sum  of  a  Column  is  10,  op  More. 

1.  A  farmer  had  3  flocks  of  sheep ;  one  of  325,  another 
436,  the  other  541 :  how  many  sheep  had  he  ? 

Analysis. — We  write  units  under  units,  tens  Operation. 

under  tens,  etc.,  and  begin  at  the  right  hand  325  8* 

as  before.    Thus,  1  unit  and  6  units  are  7  units,  436  s. 

and  5  are  12  units,  or  1  ten  and  2  units.    We  set  ^41  s. 

the  2  units  under  the  column  added,  because  it  is  

the  same  order  as  this  column,  and  add  the  1  ten  -c,■/'0•  xov  °« 
to  the  next  column,  because  it  is  the  same  order 
as  that  column.  Now  1  ten  added  to  4  tens  makes  5  tens,  and  3 
are  8  tens,  and  2  are  10  tens,  or  1  hundred  and  o  tens.  \V«  sot 
the  o,  or  right  hand  figure,  under  the  column  added,  because 
there  are  no  units  of  this  order,  and  add  the  1  hundred  to  the 
next  column,  because  it  is  the  same  order  as  that  column.  Add- 
ing  the  1  hundred  to  the  next  column,  the  sum  is  13  hundred, 
or  1  thousand  and  3  hundred.  This  being  the  last  column,  we 
set  down  the  whole  sum,  putting  the  3  under  the  column  added,  be- 
cause it  is  the  same  order  as  this  column,  and  the  1  in  the  next,  01 
thousands  place,  because  it  is  thousands.    Therefore,  etc. 


ADDITION. 


27 


10.  When  the  sum  of  a  column  exceeds  9,  what  do  you  do  ? 
Write  the  units  figure  under  the  column,  and  add  the 

tens  to  the  next  higher  order. 

11.  What  do  you  do  with  the  last  column? 
Set  down  the  whole  sum. 

-  12.  What  is  adding  the  tens  to  the  next  order  called  ? 
Carrying  the  tens. 


Thou. 

HUND. 

Tens. 

Units. 

Ill 

11 

Hill 

llll 

III 

llllll 

Mill 

llll 

1 

M. 

H. 

T. 

A 

.m't  1 

Ill 

0 

li 

Carrying  Illustrated  by  Unit  Marks. 

Analysis. — 325  =  3  hun- 
dred+2  tens +5  units;  436 
=  4  hundred  +  3  tens  +  6 
units;  and  541  =  5  hundred 
+  4  tens  + 1  unit.  Now,  to 
represent  the  first  number, 
we  place  three  counters  in 
the  column  of  hundreds,  two 
in  the  column  of  tens,  and  Jive  in  the  column  of  units.  The  other 
numbers  are  represented  in  a  similar  manner.  Beginning  at  the 
right  hand,  we  find  there  are  12  counters  in  units  column.  With- 
drawing ten  of  them,  two  will  be  left,  which  we  put  under  the 
column  added.  Now  10  units  make  1  ten ;  hence,  to  represent 
the  ten  units  withdrawn,  we  put  a  single  counter  (T.),  denoting 
a  unit  of  the  2d  order,  in  the  column  of  tens.  Again,  adding  the 
column  of  tens,  we  find  there  are  ten  counters,  and  withd.-L.wing 
ten  tens  from  this  number,  no  tens  are  left.  To  express  the 
absence  of  tens,  we  put  a  cipher  in  tens  place.  But  ten  tens 
make  1  hundred ;  hence,  to  represent  the  ten  tens  withdrawn, 
we  put  a  single  counter  (H.),  denoting  a  unit  of  the  3d  order, 
in  the  column  of  hundreds.  Finally,  in  the  hundreds  column, 
there  are  13  counters,  and  withdrawing  ten  of  them  leaves  three. 
We  therefore  put  three  counters  in  the  column  of  hundreds,  and 
to  represent  the  ten  withdrawn,  we  put  a  single  counter  (M.),  de- 
noting a  unit  of  the  4th  order,  in  the  column  of  thousands.  The 
amount  is  1  thousand,  3  hundred,  o  tens,  and  2  units.  Hence, 
carrying  the  tens  is  simply  taking  a  part  from  a  lower  order  and 
adding  it  to  the  next  higher,  which  can  no  more  affect  the  amount 
than  it  will  affect  the  amount  of  money  a  man  has,  if  he  changes 
10  cents  for  a  dime. 


28 


ADDITION. 

M 

(30 

(4.) 

(50 

(6.) 

4683 

3605 

5072 

7304 

8097 

3427 

7036 

38l6 

6og4 

3845 

5°32 

5783 

730I 

3752 

5132 

13.  The  preceding  principles  may  be  summed  up  in 
the  following 

GENERAL    RULE. 

I.   Place  the  numbers  one  under  another,  units  under 
units,  etc.,  and  beginning  at  the  right,  add  each  column 


II.  If  the  sum  of  a  column  does  not  exceed  nine, 
write  it  under  the  column  added. 

If  the  sum  exceeds  nine,  write  the  units  figure  under 
the  column,  and  carry  the  tens  to  the  next  higher  order. 

Finally,  set  down  the  whole  sum  of  the  last  column. 

Proof. — Begin  at  the  top  and  add  each  column  down- 
ward.   If  the  two  results  agree,  the  work  is  right. 

Note. — This  method  of  proof  depends  upon  the  principle  that 
reversing  the  order  of  the  figures  will  be  likely  to  detect  any  error 
that  may  have  occurred  in  the  operation.  The  learner  should 
prove  every  answer. 

EXAMPLES    FOR    PRACTICE. 


(».) 

(2.) 

(3.) 

(4.) 

(5.) 

Yards. 

Pounda 

Rods. 

Dollars. 

Acres. 

135 

333 

496 

542 

604 

26$ 

664 

175 

37 

160 

786 

548 

586 

764 

489 

182 

345 

257 

343 

853 

348 

563 

845 

577 

348 

ADDITION. 

(6.) 

(7-) 

(8.) 

(9.) 

(10.) 

684 

103 

496 

840 

965 

937 

85 

37 

6 

4 

685 

967 

4 

28 

382 

129 

49 

132 

394 

4i 

845 

732 

563 

825 

985 

29 


11.  Herbert  read  235  pages  one  day,  264  the  next, 
*nd  362  the  next :  how  many  pages  did  he  read  in  all  ? 

12.  What  is  the  sum  of  2362  days,  375  days,  and  27 
days? 

13.  If  a  yoke  of  oxen  cost  250  dollars,  and  a  cart  119 
dollars,  what  will  both  cost  ? 

14.  A  man  paid  67  dollars  for  his  coat,  16  for  his  vest, 
23  for  his  pants,  and  13  dollars  for  his  boots:  what  did 
he  pay  for  his  suit  ? 

1 5.  A  farmer  has  four  flocks  of  sheep,  one  flock  con- 
taining 256  sheep,  the  second  320,  the  third  195,  and 
the  fourth  168:  how  many  sheep  had  he  ? 

16.  What  is  the  sum  of  five  hundred  and  sixty-one, 
two  hundred  and  seven,  and  nine  hundred  and  fourteen? 

1 7.  What  is  the  sum  of  twelve  thousand  and  twelve, 
six  thousand  and  two,  and  ninety-five  hundred  ? 

18.  A  man  paid  2250  dollars  for  his  farm,  1600  dol- 
lars for  stock,  and  had  168  dollars  left:  how  much 
money  had  he  at  first  ? 

19.  Washington  was  born  in  the  year  1732,  and  lived 
67  years:  what  year  did  he  die  ? 

20.  A  man  paid  6270  dollars  for  a  horse  and  sold  it 
for  1565  dollars  more  than  he  paid  for  it:  how  much 
did  he  get  for  it? 

21.  In  what  year  will  a  person  born  in  1865,  be  21 
years  old  ? 


30  ADDITION. 

22.  What  is  the  sum  of  643  yards,  820  yards,  605 
yards,  and  319  yards? 

23.  If  I  pay  925  dollars  for  house  rent,  430  dollars  for 
clothing,  and  768  dollars  for  other  expenses,  how  much 
shall  I  spend  in  a  year  ? 

24.  The  age  of  four  brothers  is  89,  84,  78,  and  67 
years  respectively :  what  is  their  united  age  ? 

25.  John  has  63  marbles,  Henry  41,  and  William  as 
many  as  both  the  others :  how  many  did  they  all  have  ? 

26.  What  is  the  sum  of  453  dols.,  269  dols.,  804  dols., 
1000  dols.  ? 

27.  In  a  certain  army  there  are  28260  infantry,  16325 
cavalry,  and  1328  artillery :  how  many  men  did  the  army 
contain  ? 

28.  A  man  bequeathed  his  wife  23260  dols.,  his  son 
17380  dols.,  and  his  daughter  the  same  as  his  son :  how 
much  did  he  leave  them  all  ? 

29.  What  is  the  sum  of  365  days,  873  days,  219  days, 
and  35  days? 

30.  If  a  vessel  sails  235  miles  a  day  on  three  succes- 
sive days,  how  far  will  she  be  from  port  ? 

31.  My  neighbor's  farm  contains  563  acres,  and  my 
own  435  acres :  how  many  acres  do  they  both  contain  ? 

32.  Eequired  the  sum  of  1725  years  +  1007  years  -+- 
8520  years. 

^.  Required  the  sum  of  1308  ounces  +  710  ounces 
+  353  ounces  and  42  ounces. 

34.  Required  the  sum  of  2103  pounds  +  106  pounds 
+  26  pounds  +  89  pounds  +  645  pounds. 

35.  If  a  man  annually  receives  1350  dollars  salary, 
and  350  dollars  interest,  what  is  his  annual  income  ? 

36.  A  man  lias  four  farms ;  one  containing  340  acres, 
another  235  acres,  another  250  acres,  and  the  other  178 
acres :  how  many  acres  were  there  in  all  ? 


ADDITION. 

3: 

(370   (38.) 

(390 

(40.) 

(4i.) 

(42.) 

Dols.   Dols. 

Dols. 

Dols.  Cts. 

Dole.  Cts. 

Dols.  Cts. 

35    82 

423 

3 

45 

. 

4  26 

5  75 

42    40 

607 

2 

76 

, 

3  75 

3  81 

37    61 

440 

6 

08 

. 

4  9° 

1  09 

61    43 

851 

4 

30 

; 

5  7i 

6  75 

70    17 

760 

7 

05 

; 

2  43 

2  33 

25    28 

978 

8 

26 

, 

3  78 

8  45 

38    73 

465 

7 

40 

1 

9  25 

9  67 

47    68 

886 

2 

61 

, 

4  08 

7  30 

63    94 

529 

8 

35 

1 

6  25 

8  05 

85    87 

735 

3 

42 

' 

4  16 

9  35 

(430 

(44.) 

(450 

(46.) 

(470 

Dols.  Cts.   Dols. 

Cts. 

Dols. 

Cts. 

Dols. 

Cts. 

Dols.  Cts. 

29  13 

40 

53 

49 

31 

52 

60 

467  53 

3°  5° 

24 

47 

12 

53 

29 

35 

613  64 

42  35 

63 

15 

64 

31 

42 

18 

87  02 

10  78 

70 

40 

52 

49 

60 

20 

75*  34 

2<?  42 

34 

^5 

38 

24 

73 

00 

872  60 

72  96 

62 

12 

63 

19 

28 

67 

493  °4 

26  04 

73 

68 

72 

43 

49 

28 

81  73 

50  30 

58 

76 

30 

04 

83 

00 

757  02 

29  l7 

82 

94 

72 

85 

16 

27 

563  4o 

36   23 

64 

47 

37 

23 

40 

23 

80  33 

47  58 

87 

28 

52 

92 

7i 

19 

1  94 

53  22 

92 

86 

46 

25 

83 

24 

693  03 

84  35 

73 

52 

56 

83 

85 

^ 

903  48 

***  Exercises  in  adding  long  columns  upon  the  slate  or  black, 
board  are  highly  useful  in  acquiring  accuracy  and  rapidity,  and 
should  be  supplemented  by  the  teacher.  Columns  of  single  figures 
are  preferable  for  beginners. 


SUBTRACTION 


MENTAL     EXERCISES. 

To  Teachers.— The  design  of  this  Exercise  is  to  teach  the  pupil  how  to 
illustrate  the  'process  and  the  result  of  taking  one  number  from  another. 

i.  If  you  have  2  apples,  and  a  boy  takes  1  of  them 
away,  how  many  will  you  have  left  ? 

"  One  apple  taken  from  2  apples  leaves  1  apple." 

2.  Show  this  by  your  fingers  or  unit  marks. 

3.  If  you  have  3  peaches,  and  give  1  of  them  to  your 
sister,  how  many  will  you  then  have  ?     Show  this. 

4.  If  you  have  5  cents,  and  lose  2  of  them,  how  many 
will  you  have  ?     Show  this. 

5.  George  had  4  pears,  and  sold  2  of  them :  how  many 
did  he  then  have  ?     Show  this. 

6.  Jennie  had  6  roses,  and  gave  3  of  them  to  her 
teacher:  how  many  had  she  left ?    Show  this. 

7.  James  had  6  cents,  and  spent  4  of  them  for  candy: 
how  many  cents  did  he  have  left  ?     Show  this. 

8.  A  schoolboy  had  10  marbles,  and  lost  5  of  tliem: 
how  many  had  he  left  ?     Show  this. 

9.  If  you  buy  a  slate  for  8  cents,  and  sell  it  for  5  cents, 
how  much  will  you  lose  ?    Show  this. 

10.  William  gained  7  credit-marks,  but  lost  3  by  bad 
conduct :  how  many  did  he  then  have  ? 

1 1.  The  price  of  a  hat  is  6  dollars,  and  that  of  a  cap  2 
dollars :  what  is  the  difference  in  their  price  ? 

12.  In  a  certain  class  there  are  9  girls  and  6  boys : 
how  many  more  girls  than  boys  in  the  class  ? 

13.  If  you  pay  5  cents  for  an  orange,  and  sell  it  for  10 
cents,  how  much  will  you  gain  ? 

14.  4  from  10  leaves  how  many  ?    4  from  12  ? 


SUBTRACTION. 


33 


SUBTRACTION     TABLE. 


2  from 

2 

;  from 

4  from 

5  from 

2 

leaves 

0 

3 

leaves 

O 

4  leaves 

0 

5  leaves  0 

3 

a 

I 

4 

tit 

I 

5       " 

1 

6       u       1 

4 

a 

2 

5 

a 

2 

6       " 

2 

7       "       2 

5 

u 

3 

6 

a 

3 

7       " 

3 

8       "       3 

6 

ii 

4 

7 

a 

4 

8       " 

4 

9       "       4  I 

7 

a 

5 

8 

a 

5 

9       * 

5 

10       "       5 

8 

a 

6 

9 

a 

6 

10       " 

6 

11       "       6 

9 

a 

7 

10 

a 

7 

11       " 

7 

12       "       7 

i 

IO 

a 

8 

11 

a 

8 

12       " 

8 

13       "       8 

ii 

a 

9 

12 

a 

9 

13       " 

9 

14       "       9 

12 

a 

10 

13 

a 

10 

14       " 

10 

15       "      10 

6  from 

7  from 

8  from 

9  from 

6 

leaves 

0 

7 

leaves 

0 

8  leaves 

0 

9  leaves  0 

7 

u 

1 

8 

a 

1 

9       " 

1 

10       "       1 

8 

ii 

2 

9 

n 

2 

10       " 

2 

11       "       2 

9 

ii 

3 

10 

a 

3 

11       " 

3 

12       "       3 

IO 

it 

4 

11 

a 

4 

12       " 

4 

13       "       4 

ii 

ii 

5 

12 

a 

5 

13 

5 

H      "       5 

12 

li 

6 

13 

it 

6 

14       " 

6 

15       "       6 

13 

ii 

7 

14 

a 

7 

15       « 

7 

16       "       7 

14 

ii 

8 

15 

a 

8 

16       " 

8 

17       "        8 

15 

ii 

9 

16 

tt 

9 

17       " 

9 

18       "       9 

16 

ii 

10 

17 

a 

10 

18       " 

10 

19       "      10 

1.  Show  by  counters  or  unit  marks  how  many  2  takea 
from  3  will  leave  ? 

2.  Show  how  many  2  taken  from  4  will  leave. 

3.  Show  "how  many  2  taken  from  5  will  leave. 

4.  Show  how  many  3  taken  from  7  will  leave. 

5.  Show  how  many  3  taken  from  8  will  leave,  etc. 


Note. — It  is  advisable  to  let  young  pupils  verify  the  resvlts  of 
the  Subtraction  Table  by  counters  or  unit  marks,  before  they  are 
required  to  commit  it  to  memory. 


34  SUBTRACTIOK. 

DEFINITIONS. 

1 .  What  is  Subtraction  ? 

Subtraction  is  taking  one  number  from  another, 

2.  What  is  the  number  to  be  subtracted  called  ? 

The  Subtrahend. 

3.  The  number  from  which  the  subtraction  is  made  ? 

The  Minuend. 

4.  What  is  the  number  obtained  by  subtraction  called  ? 

The  Difference,  or  remainder. 

i.  When  it  is  said  that  5  taken  from  9  leaves  4,  which 
is  the  minuend  ?     The  subtrahend  ?     The  remainder  ? 

2.  When  it  is  said  that  6  taken  from  14  leaves  8,  what 
is  the  6  called  ?     The  14  ?     The  8  ? 

5.  When  both  numbers  are  the  same  denomination,  what  is 
the  operation  called  ? 

Simple  Subtraction, 

6.  How  is  Subtraction  denoted  ? 

By  a  short  horizontal  line,  called  minus  (— ). 
When  placed  between  two  numbers,  this  sign  shows  that 
the  number  after  it  is  to  be  taken  from  the  one  before 
it.  Thus,  6  —  4,  shows  that  4  is  to  be  taken  from  6, 
and  is  read  "  6  minus  4,"  or  "  6  less  4." 

Note. — The  term  minus  is  a  Latin  word,  signifying  lest. 

Read  the  following  expressions : 
1.  12— 5  =  14  — 7.  4.  20  — 6  =  8  +  6. 

5.  50-12  =  30+8. 

6-  75  +  25  =  105  -  10  +  5c 


2-  I5~  3  =  1°  +  2- 

3-  35-i°  =  3<>  — 5- 


MENTAL    EXERCISES. 

1.  If  you  pay  9  cents  for  a  sponge,  and  sell  it  for  5 
cents,  how  much  will  you  lose  ? 

Analysis.— Five  cents  from  9  cents  leave  4  cents.    Therefore, 
you  will  lose  4  cents. 


SUBTRACTION.  35 

2.  A  farmer  having  10  cows,  sold  4  of  them:   how 
many  had  he  left  ?     6  and  what  make  ten  ? 

3.  Homer  is  1 1  years  old,  and  his  sister  5  years :  what 
is  the  difference  in  their  ages  ? 

4.  Susan  had  8  pinks,  and  gave  3  to  one  of  her  school-- 
mates :  how  many  had  she  left  ?     3  and  what  make  8  ? 

5.  If  you  have  9  doves,  and  3  of  them  fly  away,  how 
many  will  you  have  left  ? 

6.  George  having  12  dollars,  gave  4  dollars  for  a  pair 
of  skates :  how  much  had  he  left  ? 

7.  The  price  of  a  vest  is  8  dollars,  and  that  of  a  coat 
14  dollars:  what  is  the  difference  in  the  price? 

8.  A  farmer  had  13  cows,  and  sold  4  of  them:  how 
many  did  lie  then  have  ?     4  and  what  make  13  ? 

9.  Henry  is  11  years  old,  and  Charles  6  years:  what 
is  the  difference  in  their  ages  ?     6  and  what  make  11? 

10.  Count  backward  by  twos  from  50  to  o,  with  rapid- 
ity.    Thus,  fifty,  forty-eight,  forty-six,  forty-four,  etc. 

11.  Count  backward  by  threes  from  60  to  o. 

12.  Count  backward  by  fours  from  60  to  o. 

SLATE      EXERCISES. 
Copy  the  following,  and  subtracting  the  lower  num' 
ber  from  the  upper,  set  the  result  under  the  figure 
subtracted.    Eepeat  the  operation  till  it  can  be  per- 
formed without  hesitation. 


From  7 
Take    4 

8 

5 

(30 

7 
6 

(4.) 
9 

5 

(5-) 
8 
6 

(6.) 
8 

7 

(7.) 
6 
6 

(8.) 

9 

5 

(9.) 

From  8 
Take    6 

(10.) 

7 
5 

(11.) 
11 

7 

(12.) 

9 

7 

(13.) 

8 
-   8 

(i4.) 

10 

7 

(15.) 
9 

(16.) 

9 

8 

36  SUBTRACTION. 

MENTAL      EXERCISES, 
i.  John  bought  10  peaches,  and  gave  6  of  them  to 
his  brother :  how  many  did  he  have  left  ?    6  and  what 
make  io? 

2.  A  man  had  16  horses,  and  sold  7  of  them:  how 
many  had  he  left?     7  and  what  make  16  ? 

3.  Julia  solved  17  examples,  and  Harriet  8  examples: 
how  many  more  did  Julia  solve  than  Harriet  ? 

4.  8  from  1 1  leaves  how  many  ?  8  from  16  ?  8  from 
14?     8  from  15  ? 

5.  If  a  man  earns  15  dollars  a  month,  and  spends  9 
dollars,  how  many  dollars  will  he  have  left  ? 

6.  9  from  15  leaves  how  many?  9  from  17?  9 
from  14? 

7.  A  market-boy  had  18  eggs  in  his  basket,  and  let- 
ting it  fall  broke  8  of  them :  how  many  whole  ones  did 
he  have  left  ? 

8.  Frank  having  8  cents,  wishes  to  buy  a  slate  which 
costs  1 2  cents :  how  many  cents  more  does  he  need,  to 
pay  for  the  slate  ? 

9.  Count  backward  by  fives  from  70  to  o  with  rapidity. 

10.  Count  backward  by  sixes  from  72  to  o,  in  like 
manner. 

SLATE     EXERCISES. 
Copy  and  sulitract  th  3  following  as  above : 

(1.)  (2.)  (3.)  (4.)  (50  (6.)  (7.)  (8.) 
From  9  11  10  12  14  13  16  17 
Take    67879879 


(9-) 

(10.) 

(11.) 

(12.) 

(13.) 

(MO 

(15.)   (16.) 

From  14 

16 

*5 

13 

12 

16   . 

17        19 

Talce    6 

7 

8 

5 

8 

7 

1        _? 

SUBTRACTION.  37 

MENTAL    EXERCISES. 
i.  George  had  12  apples,  and  gave  5  of  them  away: 
how  many  had  he  left?     12  less  7  are  how  many  ? 

2.  Horace  is  14  years  old,  and  his  sister  is  9:  what  is 
the  difference  in  their  ages  ?     14  less  5  are  how  many  ? 

3.  A  person  having  20  acres  of  land,  sold  10  acres: 
how  much  did  he  then  have  ? 

4.  18  less  8  are  how  many  ?     17  less  5  ?     15  less  7  ? 

5.  If  from  a  piece  of  silk  containing  19  yards,  10 
yards  are  cut,  how  much  will  be  left  ? 

6.  What  is  the  difference  between  17  dols,  and  9  dols.? 

7.  If  you  pay  1 7  dollars  for  a  goat  and  sell  it  for  9 
dols.,  what  will  be  your  loss  ? 

8.  If  you  buy  a  calf  for  8  dollars,  and  sell  it  for  14 
dols.,  what  will  be  your  gain  ? 

9.  A  gardener  set  out  1 8  peach  trees,  i-nd  9  of  them 
died :  how  many  lived  ? 

10.  Count  backward  by  sevens  from  70  to  o,  as  before. 

11.  Count  backward  by  eights  from  80  to  o. 

SLATE     EXERCISES. 

When  each  Figure  in  the  Lower  Number  is  Less  than  the 
one  above  it. 

1.  What  is  the  difference  between  465  dolla: ..  and  123 
dollars  ? 

Analysis.  —  Write  the  less  number  unde~.  the  Operation. 
greater,  placing  the  units  under  units,  the  tent,  under  465  dols. 
Uns,  and  the  hundreds   under    hundreds.      Begin      123  dols. 

at  the  right,  and   proceed   thus:    3   units  from   5      

units  leave  2  units.  Set  the  2  in  units  place,  under  342  QO*S. 
the  figure  subtracted,  because  it  is  units.  Next,  2 
tens  from  6  tens  leave  4  tens.  Set  the  4  in  tens  place,  under  the 
figure  subtracted,  because  it  is  tens.  Finally,  1  hundred  from  4 
hundreds  leaves  3  hundreds.  Set  the  3  under  the  hundreds 
column  because  it  is  hundreds.     The  difference  is  342  dollars. 


6$  SUBTRACTION. 

Solve  the  following  examples  in  the  same  manner: 

(*♦)  (3.)  (4.)  (5.)  (6.) 

From       629  745  846  4382  7468 

Take        416  421  526  2150  3405 

7.  How  do  you  write  numbers  for  subtraction  ? 
Write  the  less  number  under  the  greater,  units  under 
units,  tens  under  tens,  etc. 

§.   Where  do  you  begin  to  subtract,  and  where  put  the  result  ? 

Begin  at  the  right  hand,  and  set  the  result  under  the 
figure  subtracted. 

9.  What  two  principles  are  necessary  to  be  observed  in  sub- 
traction ? 

1  st.  The  numbers  must  be  Like  numbers. 
2d.  Units  of  the    same    order  must  be  sub- 
tracted, one  from  the  other. 

Can  3  pears  be  taken  from  5  inkstands  ? 

Explain  the  reason. 

Do  3  units  from  7  tens  leave  4  units,  or  4  tens  ? 

MENTAL      EXERCISES. 

1.  10  from  16  leaves  how  many?  10  from  26?  ro 
from  46  ?     10  from  76  ?     10  from  86?     10  from  96  ? 

2.  10  from  27?  10  from  37?  10  from  57?  10 
from  47?  10  from  67?  10  from  87?  10  from  77?* 
10  from  97  ? 

3.  10  from  24?  From  35?  From  48?  From  57: 
From  63?     From  76?     From  83?    From  92? 

4.  Take  4  from  7.  4  from  17.  4  from  27.  4  from 
57.     4  from  47.     4  from  37.     4  from  67.    4  from  87. 

4  from  97.     4  from  77. 

5.  Take  5  from  8.  5  from  18.  5  from  28.  5  from 
48.      5  from  38.     5  from  58.     5  from  78.     5  from  68. 

5  from  88.     5  from  98. 


SUBTRACTION.  39 

6.  Take  6  from  19.  6  from  29.  6  from  69.  6  from 
4.9.     6  from  59.     6  from  79.     6  from  99.     6  from  89. 

7.  Take  7  from  16.  7  from  26.  7  from  46.  7  from 
2,6.  7  from  56.  7  from  76.  7  from.  66.  7  from  S6. 
7  from  96. 

8.  2  from  11.  2  from  21.  2  from  31.  2  from  41.  2 
from  51.    2  from  61.    2  from  71.    2  from  81.    2  from  91. 

9.  4  from  22.  4  from  32.  4  from  62.  4  from  52.  4 
from  42.     4  from  82.     4  from  72.     4  from  92. 

10.  5  from  13.    5  from  23.    5  from  43.    5  from  53. 

5  from  $s-    5  fr°m  63-     5  fr°m  83-    5  from  73.     5 
from  93. 

11.  6  from  23.    6  from  43.    6  from  ^3-    6  from  53.    6 

6  from  83.     6  from  93.     6  from  73. 

12.  7  from  12.     7  from  22.     7  from  52.     7  from  72. 

7  from  62.     7  from  42.     7  from  82.     7  from  92. 

13.  8  from  94.     8  from  84.     8  from  74.     8  from  64. 

8  from  54.     8  from  44.     8  from  34. 

14.  9  from  85.     9  from  75.     9  from  65.     9  from  55. 

9  from  45.     9  from  35.     9  from  25. 

15.  Count  backward  by  nines  from  90  to  o,  as  above. 

16.  Count  backward  by  tens  from  100  to  o. 

SLATE     EXERCISES. 

Copy  and  subtract  the  following  as  above : 

('•)  (»•)  (3-)  (4.) 

From     736  pounds    674  yards      8567  hats    9678  dols. 
lake      513  pounds    411  yards      4251  hats    8567  dols. 


(7-) 

(8.) 

(9-) 

(10.) 

en.) 

From    5876  in. 

6341  oz. 

7043  weeks 

8672 

9000 

Take     2314  in. 

1240  oz. 

4043  weeks 

5461 

5000 

40 


SUBTRACTION. 


MENTAL   EXERCISES. 
i.  The  age  of  a  father  is  50  years,  and  that  of  his  son 
20  years :  how  much  older  is  the  father  than  the  son  ? 

Analysis. — 50  is  5  tens,  and  20  is  2  tens ;  now  2  tens  from  5 
tens  leave  3  tens,  or  30.    Therefore,  etc. 

2.  30  from  40  leaves  how  many  ? 

3.  40  from  70  leaves  how  many  ? 

4.  24  from  47  leaves  how  many  ? 

5.  24  from  68  leaves  how  many  ? 

6.  32  from  65  leaves  how  many  ? 


20  from  40  ? 
23  from  75  ? 
35  from  47  ? 
45  from  76  ? 


SLATE    EXERCISES. 

When  a  figure  in  the  Lower  Number  is  Larger  than  the  one 

above  it. 

1.  Find  the  difference  between  723  dols.  and  476  dols.  ? 

ist  Method. — Set  down  the  numbers  and 
begin  at  the  right  hand  as  before.  Since  6 
units  cannot  be  taken  from '3  units,  we  borrow 
1  of  the  2  tens  and  add  it  to  the  3,  making 
13  units.  Now  6  from  13  leaves  7,  which  we 
set  under  the  figure  subtracted.  As  we  borrowed  1  of  the  2  tens 
there  is  but  1  left ;  and  7  tens  cannot  be  taken  from  1  ten.  We 
therefore  borrow  1  of  the  7  hundred  and  add  it  to  the  1  ten, 
making  11  tens;  and  7  from  n  leaves  4,  which  we  set  in  tens' 
place.  As  we  borrowed  1  of  the  7  hundred,  there  are  but  6  hundred 
left ;  and  4  from  6  leaves  2,  which  we  set  in  hundreds'  place. 

Borrowing  Illustrated  by  Unit  Marks. 


Operation. 

723  dols. 

476  dols. 

Ans.  247  dols. 


Hundreds. 

Tens. 

Units. 

Tens  borrowed, 

Minuend,     723=    |  |  |  1 1  \% 

Subtrahend,476=          |  |  |  | 

10  tens. 
1* 

mini 

10  units. 

Ill 

MIMI 

Remainder,  247=              |  | 

MM 

linn; 

Analysis. — Let  the  7  hundreds  of  the  minuend  be  representor! 
by  7  marks,  the  2  tens  by  2  marks,  and  the  3  units  by  3  marks 
Let  the  subtrahend  be  represented  in  like  manner. 


SUBTRACTION.  41 

Since  we  cannot  take  6  units  from  3  units,  we  borrow  one  of 
the  2  tens,  which  reduced  to  units,  Ave  add  to  the  3  units,  making 
13  units ;  and  6  from  13  leaves  7.  Next,  7  tens  cannot  be  taken 
from  1  ten  (1  ten  being  erased  and  transferred  to  the  units),  we 
therefore  borrow  one  of  the  hundreds,  and  add  it  to  the  1  ten,  making 
II  tens;  then  7  from  11  leaves  4.  Finally,  4  hundreds  from  6 
hundreds  (1  hundred  being  erased  and  transferred  to  the  tens), 
leave  2  hundred.     The  result  is  247  dollars. 

2D  Method. — As  6  units  cannot  be  taken  from  3  units,  we  add 
10  to  the  3,  making  13  ;  and  6  from  13  leaves  7,  which  we  set 
under  the  figure  subtracted.  To  balance  the  10  added  to  the  3, 
instead  of  considering  the  next  upper  figure  1  less  than  it  is, 
we  add  1  ten  to  the  7  tens,  the  next  figure  in  the  lower  number, 
making  8  tens.  But  8  tens  cannot  be  taken  from  2  tens ;  we  again 
add  10  to  the  2,  making  12  tens,  and  8  from  12  leaves  4,  which  we 
set  in  tens'  place.  Finally,  to  balance  the  10  added  to  2,  we  add  1 
to  the  next  figure  in  the  lower  number,  making  5  hundred,  and  5 
from  7  leaves  2,  which  we  set  under  the  figure  subtracted.  The 
result  is  247  dols.,  the  same  as  before. 

10.  What  is  adding  10  to  the  upper  figure  called  ? 

Borrowing  ten. 

11.  Why  does  not  borrowing  10  affect  the  difference  between 
the  two  numbers  ? 

The  First  Method  simply  transfers  a  unit  from  a 
higher  to  the  next  lower  order  of  the  minuend ;  therefore 
its  value  is  not  altered. 

By  the  Seeond  Method  the  two  numbers  are 
equally  increased  j  and  when  two  numbers  are  equally 
increased,  their  difference  is  not  altered. 

Note. — This  method  is  the  less  liable  to  mistakes,  and  is  more 
generally  practiced  by  business  men. 

1 2.  How  proceed  by  the  second  method,  when  the  figure  in  the 
lower  number  is  larger  than  the  one  above  it  ? 

Add  10  to  the  upper  figure,  then  subtract,  and  add  i 
to  the  next  figure  in  the  lower  number. 


43  SUBTRACTION. 

(2.)  (3-)  (40  (50  (6.) 

From    4363  5830  7406  8738  9847 

Take     2172  3517  5183  7329  8043 

15.  The  preceding  principles  may  be  summed  up  in 
the  following 

GENERAL  RULE. 

I.  Place  the  less  number  under  the  greater,  units  under 
units,  tens  under  tens,  etc. 

II.  Begin  at  the  right,  and  subtract  each  figure  in  the 
lower  number  from  the  one  above  it,  setting  the  remainder 
under  the  figure  subtracted. 

III.  If  a  figure  in  the  lower  number  is  larger  than  the 
one  above  it,  add  10  to  the  upper  figure  ;  then  subtract, 
and  add  1  to  the  next  figure  in  the  lower  number. 

Proof. — Add  the  remainder  to  the  subtrahend;  if  the 
sum  is  equal  to  the  minuend,  the  work  is  right. 

Note. — This  proof  depends  upon  the  Axiom  that  the  whole  is 
equal  to  the  sum  of  all  its  parts. 


EXAMPLES    FOR 

PRACTICE. 

(I.) 

(*•) 

(30 

(4-) 

From 

465 

6253 

7464 

629O 

Take 

230 

3145 

4273 

6146 

(50 

(6.) 

(70 

(8.) 

From 

5434 

8670 

7202 

629O 

Take 

4260 

3452 

4101 

4062 

9.  From  6435  quarts,  take  4268  quarts. 

10.  From  265045  barrels,  take  120328  barrels. 


SUBTRACTION.  43 

ii.  A  farmer  having  2568  bushels  of  corn,  sold  1830 
bushels :  how  many  bushels  had  he  left  ? 

12.  A?s  income  is  2345  dollars,  B's  3068  dollars:  what 
is  the  difference  between  their  incomes  ? 

t^.  A  man  paid  1730  dollars  for  his  horses,  and  2135 
dollars  for  his  carriage :  what  was  the  difference  in  their 
cost? 

14.  What  is  the  difference  between  nineteen  hundred 
and  nine,  and  nine  hundred  and  nineteen  ? 

15.  What  is  the  difference  between  two  thousand  and 
four,  and  one  thousand  and  fourteen  ? 

16.  Find  the  difference  between  eight  hundred  and 
eight,  and  eight  thousand  and  eighty. 

17.  7800461  —  4560231.  18.  8000030  —  6234521. 
19.  7930451  —4000459.  20.  9603245  —  2896750. 
21.  6235672  —  4000563.         22.  1900000  —  899996. 

23.  If  a  farmer  has  738  sheep,  how  many  more  must 
he  buy  to  make  up  1320  ? 

24.  A  man  bought  goods  for  1943  dollars,  and  sold 
the  same  for  2365  dollars :  what  did  he  gain  : 

25.  A  man  born  in  1783,  died  in  1866:  how  old 
was  he? 

26.  A  person  bought  a  drove  of  cattle  for  5263  dollars, 
and  sold  them  for  4675  dollars:  how  much  did  he  lose ? 

27.  The  difference  between  the  ages  of  two  persons 
is  15  years,  and  the  older  is  79  years:  how  old  is  the 
younger  ? 

28.  1463  and  what  number  make  3185  ? 

29.  A  has  765  dollars,  B  1695  dollars,  and  C's  money 
was  equal  to  the  difference  between  A's  and  B's :  how 
much  money  had  C  ? 

30.  The  Pilgrim  Fathers  landed  at  Plymouth  Eock  in 
1620,  and  the  independence  of  the  colonies  was  declared 
in  1776:  how  many  years  between  these  two  events? 


44  SUBT  It  ACTION. 

DRILL    FOR    RAPID    COMBINATIONS. 

To  Tbacheks.— These  exercises,  if  properly  conducted,  will  secure  tw< 
objects :  First,  the  habit  of  fixing  the  attention  ;  Second,  rapidity  in  the  com- 
bination of  numbers.  They  should  be  dictated  slowly  at  first,  increasing  hi 
speed  as  the  class  acquire  ability  to  follow.  The  answers  may  be  given 
individually,  or  by  the  class  simultaneously. 

!  Oral. — i.  From  12,  subtract  5;  add  6;  subtract  4; 
add  3 ;  add  4 ;  subtract  10 ;  add  9 ;  subtract  4 ;  add  2  ; 
subtract  5  :    what  is  the  result  ? 

Explanation. — The  teacher  says,  "  from  12  subtract  5,"  the 
class  think  7 ;  "  add  6,"  the  class  think  13  ;  "  subtract  4,"  the  class 
think  9 ;  "  add  3,"  the  class  think  12,  *aid  so  on. 

2.  To  9,  add  4 ;  subtract  2 ;  add  5 ;  subtract  6 ;  add 
3 ;  subtract  5  ;  add  7  ;  add  3  ;  mbtiact  5  :  the  result  ? 

3.  From  17  take  15 ;  add  10 ;  take  2  ,  add  9;  add  6 ; 
take  5  ;  take  10 ;  add  7  ;  take  2  :  result  ?    • 

4.  To  23  add  5  ;  take  6 ;  add  9 ;  add  4 ;  take  10 ;  add 
9;  take  4;  add  7  ;  add  10:  take  9;  add  4:  result  ? 

5.  From  24  take  6;  add  8;  take  10;  add  5;  add  7; 
take  3 ;  add  7  ;  take  6  ;  add  5  ;  add  3  :  result  ? 

6.  To  35  add  4 ;  take  6 ;  take  3 ;  add  8 ;  take  7 ;  add 
5  ;  add  3 ;  take  9 ;  add  7  ;  take  8  ;  add  10  :  result  ? 

Slate.— 1.  To  375  add  123;  subtract  47;  add  23; 
add  47  ;  subtract  36  ;  add  87  ;  subtract  68  :  the  result  ? 
5  2.  From  62  take  34  ;  add  76  ;  take  40 ;  add  78  ;  take 
99  ;  add  76  ;  add  24 ;  take  43  :  result  ? 

3.  Add  344  to  65 ;  take  64;  add  784;  take  678;  add 
407  ;  take  309  ;  add  860  :  result  ? 

4.  From  780  take  607  ;  add  788;  add  28;  take  19; 
add  976  ;  take  306  ;  add  1000  :  result  ? 

5.  To  4678  add  6246 ;  take  4004;  add  5020;  take  50S ; 
add  1700;  take  468;  add  2500:  result? 

6.  From  8640  take  3476;  add  4578;  take  5065;  add 
87;  take  1000;  add  608;  take  47 :  result? 


MULTIPLICATION. 


MENTAL    EXERCISES. 

To  Teachers.— The  object  of  this  Exercise  is  to  develop  the  idea  of 
h  times,'''  as  used  in  Multiplication,  preparatory  to  learning  the  Table. 

i.  If  your  father  gives  you  3  books  at  one  time,  and 
3  at  another,  how  many  books  will  you  have  ? 
"  3  books  and  3  books  are  6  books." 

2.  How  many  times  3  books  will  you  have  ? 
"  Two  times." 

3.  How  many  are  2  times  3  books  ?     "6  books." 

4.  How  many  are  2  times  2  pencils  ? 

5.  Show  this  by  counters  or  unit  marks. 

6.  How  many  are  2  cents,  and  2  cents,  and  2  cents  ? 

7.  How  many  are  3  times  2  cents  ?     Show  it. 

8.  If  you  have  4  fingers  on  each  hand,  how  many 
have  you  on  both  hands  ? 

9.  How  many  are  2  times  4  ?     Show  it. 

10.  John  has  5  apples,  and  Henry  has  2  times  as 
many :  how  many  has  Henry  ?     Show  it. 

11.  If  1  orange  costs  6  cents,  what  will  2  oranges 
cost? 

Analysis. — If  1  orange  costs  6  cents,  2  oranges  will  cost  2 
times  6  cents  ;  and  2  times  6*  cents  are  12  cents.  Therefore,  2 
oranges  will  cost  12  cents.     * 

12.  How  many  are  7  slates  and  7  slates  ?     Show  it. 

13.  If  1  yard  of  braid  costs  7  cents,  what  will  2  yards 
cost  ?     Show  it. 

14.  At  8  cents  each,  what  will  be  the  cost  of  2  ink- 
stands ?     Show  it. 

15.  If  1  writing-book  costs  9  cents,  what  will  2  writing- 
books  cost?     Show  it. 

16.  How  many  are  2  times  10  dollars  ?     Show  it 


46 


MULTIPLICATION. 


MULTIPLICATION     TABLE 


once 

2 

times 

3 

times 

4 

times 

i    is      i 

1 

are     2 

1 

are     3 

1 

are    4 

2       "          2 

2 

"       4 

2 

"      6 

2 

"       8 

3     «       3 

3 

"       6 

3 

"      9 

3 

«     12 

4     "       4 

4 

"       8 

4 

"     12 

4 

"     16 

5     "      5 

5 

"     10 

5 

"     15 

5 

"     20 

6     "       6 

'5 

"     12 

6 

"     18 

6 

"     24 

7     "       7 

7 

"     14 

7 

"     21 

7 

"     28 

8     "       8 

8 

"     16 

8 

"     24 

8 

«    32 

9     "       9 

9 

"     18 

9 

"     27 

9 

"    36 

IO       "       IO 

10 

"     20 

10 

«     30 

10 

"    40 

ii     "     ii 

11 

"     22 

n 

"     33 

n 

"     4T~ 

12       "       12 

12 

"     24 

12 

"     36 

12 

"     48 

5  times 

6  times 

7 

times 

8  times 

i    are     5 

1 

are     6 

1 

are     7 

1 

are     8 

2     "     10 

2 

"     12 

2 

"     14 

2 

"     16 

3     "     15 

3 

"     18 

3 

"     21 

3 

"     24 

4     "     20 

4 

"     24 

4 

"     28 

4 

a     32 

5     "     25 

5 

"    30 

5 

"     35 

5 

"     40 

6     "     30 

6 

«     36 

6 

"     42 

6 

*     48 

7     "     35 

7 

"    42 

7 

"     49 

7 

«     56 

8     "     40 

8 

"    48 

8 

*     5(> 

8 

"     64 

9     "     45 

9 

"     54 

9 

"     63 

9 

"     72 

.12— 1LJ° 

10 

"     60 

10 

"     70 

10 

t(       oQ 

^i"3s— 

"     66 

n 

"     77 

11 

«     88 

12     "     60 

12 

"     72 

12 

"     84 

12 

"     96 

9  times 

IC 

times 

II 

times 

12 

times 

1    are     9 

I 

are  10 

1 

are  n 

1 

are  12 

2     "     18 

2 

"     20 

2 

"     22 

2 

"     24 

3     "     27 

3 

"    30 

3 

"     33 

3 

"    36 

4     "     36 

4 

"    40 

4 

"     44 

4 

«    48 

5     "   .45 

5 

"    50 

5 

"     55 

5 

"    60 

6     "     54 

6 

"    60 

6 

«     66 

6 

u     72 

7     "     63 

7 

"     70 

7 

"     77 

7 

"    84 

8     «     72 

8 

"    80 

8 

«     88 

8 

"    96 

9     "     81 

9 

"    90 

9 

"     99 

9 

"  108 

10     "     90 

10 

"  100 

10 

"  no 

10 

"  120 

1 1     "     99 

11 

"  no 

11 

"  121 

n 

"  132 

12     "   108 

12 

"  120 

12 

"  132 

12 

*  i44 

MULTIPLICATION.  4? 

DEFINITIONS. 

1 .  What  is  Multiplication  ? 

Multiplication  is  finding  the  amount  of  a  num- 
ber taken  or  added  to  itself  a  given  number  of  times. 

2.  What  is  the  number  to  be  multiplied  called? 

The  Multiplicand, 

3.  What  the  number  by  which  you  multiply  ? 

The  Multiplier  ;  and  shows  how  many  times  the 
multiplicand  is  to  be  taken. 

4.  What  is  the  number  obtained  by  multiplication  called  ? 

The  Product. 

When  it  is  said  that  3  times  4  are  1 2,  which  is  the 
multiplicand  ?     The  multiplier  ?     The  product  ? 

When  it  is  said  that  4  times  3  are  12,  what  is  the  4  ? 
The  3?     The  12? 

5.  What  else  are  the  multiplier  and  multiplicand  called  ? 
Factors ;  for  they  make  or  produce  the  product. 

The  number  12  is  made   up  of  four  3s,  or  three  4s; 
hence,  3  and  4  are  factors  of  1 2. 

Remark. — The  product  is  the  same  in  whatsoever  order  the 
factors  are  multiplied.  Thus,  if  4  be  represented  by 
a  horizontal  row  of  unit  marks  upon  the  blackboard,  **  **  ■*  * 
and  3  by  a  perpendicular  row  of  3  unit  marks,  it  is  S*  13  &  J3 
plain  that  the  horizontal  row  taken  3  times,  is  equal  ^  0  0  ^ 
to  the  perpendicular  row  taken  4  times. 

6.  When  the  multiplicand  contains  only  one  denomination, 
what  is  the  operation  called  ? 

Simple  Multip I i cation. 

7.  How  is  Multiplication  denoted? 

By  an  oblique  cross,  called  the  sign  of  multiplica- 
tion (  x ).  Thus,  6x4  shows  that  6  and  4  are  to  be 
multiplied  together,  and  is  read  "6  times  4,"  "6  into 
4,"  or  "6  multiplied  by  4." 

Read  the  following  expressions :  2x6  =  3x4;  4x5 

=  10x2;  2x2x6-12x2, 


48  MULTIPLICATION. 

MENTAL     EXERCISES. 

i.  What  will  4  pears  cost,  at  3  cents  apiece  ? 

Analysis. — Since  1  pear  costs  3  cents,  4  pears  will  cost  4  times 
3  cents ;  and  4  times  3  cents  are  12  cents.  Therefore,  4  pears  will 
cost  12  cents. 

***  It  is  important  for  the  pupil  to  analyze  every  concrete  ex- 
ample in  a  concise,  distinct,  and  scholarly  manner. 

2.  What  will  5  oranges  cost,  at  6  cents  apiece  ? 

3.  If  1  hat  costs  6  dollars,  what  will  4  hats  cost  ? 

4.  At  7  dollars  a  barrel,  how  much  will  3  barrels  of 
flour  come  to  ? 

5.  If  you  obtain  6  credit-marks  each  da)7  for  5  days  in 
succession,  how  many  will  you  have  ? 

6.  In  1  week  there  are  seven  days :  how  many  days 
are  there  in  6  weeks  ? 

7.  George  has  7  marbles,  and  Henry  has  4  times  as 
many :  how  many  marbles  has  Henry  ? 

8.  At  8  cents  apiece,  what  will  6  tops  cost  ? 

9.  At  9  dollars  each,  what  will  4  trunks  cost  ? 

SLATE     EXERCISES. 

Copy  and  multiply  the  following,  setting  each  result 
nnder  the  figure  multiplied : 

(1.)     (2.)       (3.)       (4.)       (50       (6.)       (7.)     (8.) 
Mult.    67897589 

ByAAAA     —     1     —     2. 

Prod.  30  45  6$ 

(9.)      (10.)     (11.)     (12.)     (13.)     (14.)    (15.)  (16.) 

Mult.  8  9  7  9  8  9         10         11 

2fy     78969789 


MULTIPLICATION.  49 

MENTAL    EXERCISES. 

i.  Bought  7  barrels  of  flour,  at  6  dollars  a  barrel: 
what  did  the  flour  come  to  ? 

2.  Sold  8  silk  umbrellas,  at  7  dollars  each :  what  was 
the  amount  of  the  bill  ? 

3.  If  a  man  earns  9  dollars  a  week,  how  much  will  he 
earn  in  8  weeks  ? 

4.  What  must  be  paid  for  6  quarts  of  cherries,  at  12 
cents  a  quart  ? 

5.  In  a  certain  orchard  there  are  8  rows  of  trees,  and 
12  trees  in  a  row:  how  many  trees  does  it  contain ? 

6.  If  1  table  costs  8  dollars,  what  will  be  the  cost  of 
10  tables  ? 

7.  What  must  I  pay  for  11  yards  of  muslin,  which  is 
12  cents  a  yard? 

8.  How  many  quarts  in  1 1  pecks,  allowing  8  quarts 
to  a  peck  ? 

9.  In  1  dime  there  are  10  cents:  how  many  cents  are 
there  in  1 1  dimes  ? 

10.  In  1  year  there  are  12  months:  how  many  months 
are  there  in  10  years? 

SLATE     EXERCISES. 

When  the  Multiplier  has  but  one  figure,  and  the  Product  of 
each  figure  in  the  Multiplicand  is  Less  than  10. 

1.  Multiply  1232  by  3. 

Analysis. — Write  the  multiplier  under  the  mul-  Operation. 

fciplicand,    and   begin    at  the    right,      3    times    2  1232 
units  are  6  units.     Set  the  6  in  units  place,  under  ~ 

the  figure  multiplied,  because  it  is  units.     3  times  

3  tens  are  9  tens.     Set  the  9  in  tens  place,  because  3696  Am, 
it  is  tens.     3  times  2  hundreds  are  6  hundreds.     Set 
the  6  in  hundreds  place,  because,  etc.     3  times  1  thousand  are  3 
thousands.     Set  the  3  in  thousands  place. 


50  MULTIPLICATION. 

Copy  and  multiply  the  following  in  like  manner: 

(«•)  (3-)  (4-)  (5-) 

Mult.      42414  22321  12212  iiiii 

By  2  3  4  r 


(6.) 

(70 

(8.) 

(9.) 

Mult. 
By 

IOIIOI 

6 

332032 
3 

IIOIOI 

7 

111111 
8 

MENTAL    EXERCISES, 
i.  At  9  dollars  a  barrel,  what  will  8  barrels  of  cran- 
berries come  to  ? 

2.  What  will  be  the  cost  of  6  flutes,  at  12  dollars  each  ? 

3.  A  farmer  sold  1 1  calves,  at  9  dollars  apiece :  what 
did  he  receive  for  them  ? 

4.  How  many  are  9  times  7  ?    8  times  9  ? 

5.  Bought  10  accordions,  at  12  dollars  each :  what  was 
the  amount  of  the  bill  ? 

6.  How  many  are  8  times  7  ?     9  times  8  ? 

7.  If  1  plough  cost  11  dollars,  what  will  be  the  cost 
of  12  ploughs,  at  the  same  rate  ? 

8.  How  many  are  11  times  10?     12  times  11  ? 

WRITTEN     EXERCISES. 
When  the  Product  of  the  respective  figures  is  10,  or  More. 

1.  If  1  horse  costs  435  dollars,  how  much  will  3 
horses  cost? 

Analysis. — Since  1  horse  costs  435  dollars,  Operation. 

3  horses  will  cost  3  times  as  much.     Write  the  435  mult'd. 

multiplier  under  the  multiplicand,  and  bepinnintf  -  mult 

at  the  right,  proceed  thus :  3  times  5  units  are     

15  units;  we  set  the  5  in  units  place,  under  the  I3°5  dols. 

figure  by  which  we  multiply,  and  carry  the  1  to 

the  product  of  the  nt'^t  figure,  as  in  addition.   Next, 3  times  3  tens 


MULTIPLICATION.  51 

are  9  ten?,  and  1  (to  carry)  makes  10  tens  ;  we  set  the  o  in  tens 
place,  and  carry  the  1  to  the  product  of  the  next  figure.  Finally, 
3  times  4  hundred  are  12  hundred,  and  1  (to  carry)  makes  13  hun- 
dred.    Therefore,  3  horses  will  cost  1305  dollars. 

7.  How  write  numbers  for  multiplication? 

Write  the  multiplier  under  the  multiplicand,  units 
under  units,  etc. 

§.  How  proceed  when  the  multiplier  contains  but  one  figure  ? 

Begin  at  the  right  hand,  and  multijoly  each  figure  in 
tlie  multiplicand  by  the  multiplier,  separately. 

9.  What  do  you  do  with  the  partial  results,  when  10,  or 
more  ? 

Set  the  units  figure  under  the  figure  multiplied,  and 
carry  the  tens  to  the  product  of  the  next  figure. 

10.  When  the  multiplier  is  units,  what  order  is  the  product  ? 
The  same  order  as  the  figure  multiplied. 

11.  What  are  the  principles  as  to  the  nature  of  the  multiplier, 
the  multiplicand,  and  the  product  ? 

1st.  The  J&lultipli&r  must  always  be  considered  an 
abstract  number. 

2d.  The  Multiplicand  may  be  an  abstract,  or  con- 
crete  number. 

3d.  The  Product  is  always  the  same  name  or  hind 
as  the  true  multiplicand;  for,  repeating  a  number  does 
not  change  its  nature. 

12.  Which  of  the  factors  is  the  true  multiplicand? 

The  true  multiplicand  is  that  number,  which 
added  to  itself  the  given  number  of  times,  will  produce 
the  required  product. 

Remark. — Neither  a  concrete  nor  abstract  number  can  properly 
be  said  to  be  repeated  as  many  times  as  another  is  long,  or  heavy. 
Hence,  money  can  not  be  multiplied  by  yards,  pounds,  etc. ;  but 
any  given  sum  can  be  multiplied  by  a  number  of  units  equal  to 
the  number  of  yards,  pounds,  etc.,  in  the  given  quantity,  and  tht. 
product  will  be  money. 


62  MULTIPLICATION. 

2.  In  i  year  there  are  365  days :  how  many  days  are 
there  in  5  years  ? 

3.  If  1  piano  costs  750  dollars,  what  will  6  pianos 
cost? 

4.  If  1  farm  contains  875  acres,  how  many  acres  will 
8  farms  of  the  same  size  contain  ? 


Mult. 
By 

(50 
2136 
2 

(6.) 

7345 
3 

(7.) 

28536 
4 

(8.) 

65043 
5 

Mult 
By 

(9.) 

701230 
6 

(10.) 
635728 
7 

(11.) 
830405  • 
8 

(12.) 
973080 
9 

MENTAL    EXERCISES. 

1.  What  will  3  tables  cost,  at  45  dollars  apiece  ? 

Analysis. — 3  tables  will  cost  3  times  as  much  as  1  table.  But 
45  is  equal  to  4  tens  and  5  units.  Now  3  times  4  tens  are  12 
tens,  or  120;  3  times  5  units  are  15  uuits,  or  1  ten  and  5  units; 
and  1  ten  and  5  units  added  to  120  make  135.  Therefore,  3  tables 
will  cost  135  dollars. 

Note.— When  tie  numbers  to  be  multiplied  mentally  are  large, 
it  is  advisable  to  separate  them  into  the  units,  tens,  etc.,  of  which 
they  are  composed,  and  multiply  the  highest  order  first,  then  the 
next  lower,  etc.,  adding  the  results  as  we  proceed.    (P.  26,  N.) 

2.  How  much  can  a  man  earn  in  2  months,  if  he 
earns  36  dollars  a  month  ? 

3.  In  a  peach  orchard  there  are  5  rows  of  trees,  and 
27  trees  in  a  row:  how  many  peach  trees  does  the 
orchard  contain  ? 

4.  How  many  are  3  times  54?     4  times  37  ? 


MULTIPLICATION".  53 

5.  If  a  man  can  earn  56  dollars  a  month,  how  much 
can  he  earn  in  7  months  ? 

6.  If  1  hogshead  contains  63  gallons  of  molasses,  how 
many  gallons  will  5  hogsheads  contain  ? 

7.  If  1  melodeon  can  be  had  for  75  dollars,  what  will 
be  the  cost  of  6  melodeons  ? 

8.  If  1  sofa  is  worth  83  dollars,  how  much  are  9  sofas 
worth  ? 

WRITTEN     EXERCISES. 
When  the  Multiplier  has  two  or  more  Figures. 

1.  What  will  106  buggies  cost,  at  268  dollars  apiece? 

Analysis.— 106  buggies  will  cost  106  Operation. 
times  as  much  as  1  buggy.     We  write  ^g  rn,rilt'd 

the  multiplier  under  the  multiplicand,  as  ,         u 

1^         ^1     •     •       A  xt.     •  i  x  1     j  106  mult. 

before,  and  beginning  at  the  right  hand,  

proceed   thus:    6  times  8  units  are  48  1 608 

units.     The  8  is  set  in  units  place,  under  268     • 

the  figure  which  produced  it,  because  it  

is  units;  and  the  4  is  carried  to  the  pro-     AflS.  28408  dols. 
duct  of  the  next  figure,  because  it  is  the 

same  order  as  that  figure.  The  other  figures  of  the  multipli- 
cand are  multiplied  by  6,  and  the  results  set  down  in  a  similar 
manner.  Next,  the  product  by  o  tens  is  o ;  we  therefore  omit  it. 
Again,  1  hundred  times  8  units  are  8  hundreds.  The  8  is  set  in 
hundreds  place,  under  the  figure  which  produced  it,  because  it  is 
hundreds.  The  other  figures  of  the  multiplicand  are  multiplied 
•by  1  in  the  same  manner.  Finally,  adding  these  partial  products 
together,  the  result,  28408  dollars,  is  the  whole  product  required. 

1 3.  When  the  multiplier  has  two  or  more  figures,  how  pro- 
ceed? 

Beginning  at  the  rigid  hand,  multiply  the  multipli- 
cand by  each  figure  of  the  multiplier  separately,  and  set 
the  first  figure  of  each  partial  product  under  the  multi- 
plying figure. 


54  MULTIPLICATION. 

14.  What  js  meant  by  partial  products? 

They  are  the  several  results  which  arise  from  multi- 
plying the  multiplicand  by  the  separate  figures  of  the 
multiplier,  and  are  so  called  because  they  are  parts  of 
the  whole  product. 

1 5.  What  is  done  with  the  partial  products,  and  why  ? 

We  add  them  together,  because  the  whole  product  is 
equal  to  the  sum  of  all  its  parts. 


(*•) 

(30 

(4.) 

(5.) 

(6.) 

Mult. 

3724 

4103 

5378 

6037 

8734 

By 

25 

34 

46 

57 

78 

1 6.  The  preceding  principles  may  be  summed  up  in 
the  following 

GENERAL  RULE. 

I.  Place  the  multiplier  under  the  multiplicand,  units 
under  units,  tens  under  tens,  etc. 

II.  When  the  multiplier  has  tut  one  figure,  beginning 
at  the  right,  multiply  each  figure  of  the  multiplicand  by 
it,  and  set  down  the  result  as  in  addition. 

III.  If  the  multiplier  has  two  or  more  figures,  multiply 
the  multiplicand  by  each  figure  of  the  multiplier  sepa- 
rately, and  set  the  first  figure  of  each  partial  produd 
under  the  multiplying  figure. 

Finally,  the  sum  of  the  partial  products  will  be  the 
ansiver  required. 

Proof. — Multiply  the  multiplier  by  the  multiplicand ; 
if  this  result  agrees  with  the  first,  the  work  is  right. 

Note. — This  proof  is  based  upon  the  principle,  that  the  retail 
will  bo  the  same  whichever  of  the  given  numbers  is  taken  as  the 
multiplicand.    (P.  47,  Q.  5.) 


MULTIPLICATION. 


55 


EXAMPLES    FOR    PRACTICE. 
i.  Multiply  78  by  43,  and  prove  the  operation  ? 

Operation. 

Multiplicand    78 


Multiplier       43 


Proof. 

The  given  multiplier         4; 
"        "    multiplicand     7^ 


234 

344 

3*2 

301 

Product 

3354 

The  same  as 

the  first  335h. 

(2.) 

(30 

(4.) 

(5.) 

27356 

40256 

57189 

70203 

21 

27 

32 

47 

(6.) 

(70 

(8.) 

(9.) 

63I42O 

507060 

81367O 

973848 

158 

249 

365 

1476 

10.  There  are  24  hours  in  a  day :  how  many  hours  £re 
there  in  365  days  ? 

11.  There  are  320  rods  in  a  mile:  how  many  rods  are 
in  150  miles? 

12.  What  will  265  acres  of  land  cost  at  87  dollars  per 
acre  ? 

13.  What  cost  97  melodeons,  at  250  dollars  apiece  ? 

14.  Multiply  43846  by  123. 

15.  Multiply  57028  by  321. 

16.  Multiply  604326  by  237. 

17.  Multiply  673862  by  250. 

18.  Multiply  703562  by  304.  ' 

19.  Multiply  570031  by  402. 

20.  Multiply  439275  by  425- 
2  j.  Multiply  789426  by  521. 


56  MULTIPLICATION. 

22.  What  will  be  the  cost  of  85  pianos,  at  650  dollars 
apiece  ? 

23.  If  a  ship  sails  115  miles  in  one  day,  how  far  will 
she  sail  in  198  days? 

24.  If  there  are  63  yards  in'  1  piece  of  cloth,  how 
many  are  there  in  268  pieces  ? 

25.  At  320  dollars  a  yoke,  what  will  500  yoke  of  oxen 

30St? 

26.  What  will  no  wagons  cost,  at  175  dollars  apiece  ? 

27.  What  cost  350  suits  of  clothes,  at  115  dollars 
a  suit  ? 

28.  What  cost  1645  saddles,  at  75  dollars  apiece? 

29.  What  cost  3250  tons  of  iron,  at  87  dollars  a  ton  ? 

30.  What  does  the  President's  salary  amount  to  in  8 
years,  at  25000  dollars  a  year  ? 

31.  What  is  the  expense  of  furnishing  an  army  of 
1 1500  men  with  uniforms  which  cost  57  dollars  apiece? 

32.  If  1  ox  weighs  n 63  pounds  will  100  oxen  weigh? 
^  What  cost  465  velvet  cloaks,  at  129  dollars  apiece  ? 
34.  What  cost  1567  tons  of  lead,  at  120  dollars  per  ton? 

CONTRACTIONS. 

I.  When  the  Multiplier  is  10,  100,  1000,  etc. 

17.  What  is  the  effect  of  annexing  a  cipher  to  a  number  ? 
Annexing  one  cipher  to  a  number  multiplies  it  by  10  5 
annexing  tivo  ciphers  multiplies  it  by  100,  and  so  on. 

Remark. — The  learner  will  observe  that  each  cipher  annexed 
to  a  number,  removes  each  preceding  figure  in  the  number  to  the 
next  higher  order,  which  has  ten  times  the  value  of  the  order  from 
which  it  has  been  removed.     (Page  IX,  Q.  17.) 

2.  What  will  10  sofas  cost,  at  56  dollars  apiece? 

Solution.— ^Annexing  a  cipher  to  56  dollars,  the  result  i-»  560 
dollars,  which  is  the  cost  required. 


MULTIPLICATION.  57 

3.  What  will  100  acres  of  land  cost,  at  75  dollars 
an  acre  ? 

Solution. — Annexing  2  ciphers  to  75  dollars,  the  result  is  7500 
dollars,  which  is  the  answer  required. 

1§.  How  then  do  you  multiply  by  10,  100,  1000,  etc.  ? 

Annex  as  many  ciphers  to  the  multiplicand  as  there  are 
ciphers  in  the  multiplier,  and  the  result  ivill  be  the  pro- 
duct. 

4.  What  is  the  product  of  361  multiplied  by  100? 

5.  Multiply  453  by  100. 

6.  Multiply  2045  by  1000. 

7.  Multiply  46208  by  1000. 

8.  Multiply  58241  by  1000. 

9.  Multiply  326072  by  10000. 

10.  Multiply  4007289  by  1 00000. 

11.  What  cost  10  cows,  at  51  dollars  apiece  ? 

12.  At  265  dollars  apiece,  what  will  100  buggies 
come  to  ? 

13.  What  cost  100  acres  of  land,  at  205  dollars  per 
acre  ? 

14.  If  1  bushel  of  apples  is  worth  6^  cents,  what  will 
be  the  price  of  1000  bushels? 

II.  When  one  op  both  Factors  have  Ciphers  on  the  right. 

15.  If  1  railroad  car  costs  2700  dollars,  what  will  50 
cars  cost  ? 

Analysis. — If  1  car  cost  2700  dollars,  Operation. 

50  cars  will  cost  50  times  as  much.     We  2700 

resolve  the  multiplicand  into  the  factors  ^ 

27  and  100 ;  and  the  multiplier  into  5  and  

10.  Now,  as  the  product  is  the  same  in  Ans.  135000  dols. 
whatever  order   the   factors   are  taken, 

omitting  the  ciphers  on  the  right  of  the  multiplicand  and  multi- 
plier, we  multiply  the  significant  figures  together  as  before,  and 
annex  the  ciphers  omitted  to  the  product.   The  result  is  135000  d. 


58  MULTIPLICATION-. 

19.  How  proceed  when  one  or  both  factors  have  ciphers  on 
the  right  ? 

Multiply  the  significant  figures  together ;  and  to  the 
result  annex  as  many  ciphers  as  are  found  on  the  right 
of  both  factors. 

(16.)  (17.)  (18.) 

Mult,         i860  25000  4°53 

By  300  •  7  2000 

Prod.        558000  175000  8106000 

19.  Multiply  37000  by  31. 

20.  Multiply  52300  by  65. 

21.  Multiply  42721  by  2000. 

22.  Multiply  60045  by  3100. 

23.  Multiply  85000  by  2300. 

24.  Multiply  375000  by  57000. 

25.  Multiply  204200  by  20500. 

26.  Multiply  800400  by  600300. 

27.  What  will  200  acres-  of  land  cost,  at  70  dollars 
per  acre  ? 

28.  What  cost  21000  bushels  of  oats,  at  60  cents  a 
bushel  ? 

29.  If  a  man  travels  120  miles  a  day,  how  far  can  he 
travel  in  300  days  ? 

30.  If  1  acre  produces  50  bushels  of  corn,  what  will 
3000  acres  produce  ? 

31.  Multiply  25  thousand  by  25  hundred. 

32.  Multiply  two  hundred  and  forty-five  thousand  by 
16  thousand. 

33.  Multiply  65  thousand  and  seventy  by  21  thou- 
sand seven  hundred. 

34.  Multiply  one  million,  one  hundred  and  ten  thou- 
sand, by  26  thousand. 


MULTIPLICATION.  59 

QUESTIONS     FOR     REVIEW. 

Oral.— i.  If  9  men  can  bnild  a  wall  in  12  days,  how 
long  will  it  take  1  man  to  build  it  ? 

Analysis. — It  will  take  1  man  9  times  as  long  as  9  men,  and 
9  times  12  days  are  108  days.  Therefore,  it  will  take  I  man  108 
days. 

2.  If  a  jar  of  butter  will  last  a  family  of  8  persons  6 
weeks,  how  long  will  it  last  1  person  ? 

3.  Henry  can  read  a  book  through  in  1 1  days  by  read- 
ing 6  hours  each  day :  how  long  will  it  take  him  if  he 
reads  1  hour  a  day  ? 

4.  If  12  men  can  frame  a  house  in  8  days,  how  long 
will  it  take  1  man  to  frame  it  ? 

5.  If  I  buy  4  barrels  of  apples  at  3  dollars  a  barrel, 
and  4  barrels  of  pears  at  5  dollars,  what  will  be  the  cost 
of  both  ? 

6.  A  farmer  haying  15  bushels  of  wheat,  sold  9  bush- 
els at  2  dollars  a  bushel,  and  the  remainder  at  3  dollars 
a  bushel :  how  much  did  he  get  for  his  wheat  ? 

Written. — 1.  If  it  takes  285  laborers  18  months 
to  build  a  railroad,  how  long  would  it  take  1  man  to 
build  it  ? 

2.  A  ship  of  war  has  provisions  to  last  a  crew  of  625 
men  90  days :  how  long  would  they  last  1  man  ? 

3.  If  a  clerk  has  36  dollars  a  month  for  the  first  4 
months ;  48  dollars  a  month  for  the  next  4 ;  and  6c 
dollars  a  month  for  the  next  4 ;  what  will  he  receive  for 
the  year  ? 

4.  A  man  having  1000  dollars  in  his  pocket,  gave  45 
dollars  each  to  1 2  poor  persons :  how  much  had  he  left  ? 

5.  If  I  receive  150  dollars  a  month,  how  much  shall  I 
have  at  the  end  of  the  year,  after  deducting  28  dollars  a 
month  for  board? 


DIVISION. 


MENTAL     EXERCISES. 

10  TEA.CHBHS.— The  object  of  this  preliminary  Exercise  is  to  develop 
Ute  idea  of  "  times,'"  as  used  in  Division,  preparatory  to  learning  the  Table. 

i.  If  I  have  9  pencils,  how  many  boys  can  I  supply 
with  3  pencils  each  ? 

Analysis. — If  I  give  one  boy  3  pencils,  how  many  will  be  left  ? 

"  Six  pencils." 

If  I  give  another  boy  3,  how  many   pencils  will   be   left? 

"Three." 

If  I  give  another  3,  how  many  will  be  left  ?    "  None." 
How  many  boys  have  I  supplied  with  3  pencils  ?     "  Three." 
How  many  times  are  3  pencils  contained  in  9  pencils? 
"  Three  times." 

2.  How  many  peaches,  at  2  cents  each,  can  you  buy 
for  8  cents  ?     Show  this  by  counters. 

3.  How  many  oranges,  at  4  cents  each,  can  you  buy  for 
12  cents?     How  many  times  4  make  12 ?     Show  this. 

4.  In  1  gallon  there  are  4  quarts:  how  many  gallons 
are  there  in  8  quarts  ?  How  many  times  4  make  8  ? 
Show  this  by  unit  marks. 

5.  If  a  lad  earns  5  dollars  a  week,  how  long  will  it  take 
him  to  earn  25  dollars?  How  many  times  5  make  25  ? 
Show  this. 

6.  At  6  cents  an  ounce,  how  many  ounces  of  candy 
can  you  buy  for  18  cents?    Show  this  by  unit  marks. 

7.  How  many  lambs,  at  2  dollars  apiece,  can  be  bad 
for  20  dollars  ?     Show  this. 

8.  At  4  dollars  a  pair,  how  many  pair  of  boots  can  I 
buy  for  16  dollars?     Show  this. 

9.  If  I  have  20  pounds  of  flour,  how  many  poor  per- 
sons can  I  supply  with  5  pounds  each  ? 


Division. 


61 


DIVISION     TABLE. 


i  is  in 

2  is  in 

3  is  in 

4  is  in 

i,  once.* 

2,  once. 

3,  once. 

4,  once. 

2,          2 

4,          2 

6,          2 

8,          2 

3,         3 

6,          3 

9,          3 

12,          3 

4,          4 

8,          4  , 

12,          4 

16,          4 

5,          5 

10,          5 

15,          5 

20,          5 

6,          6 

12,          6 

18,          6 

24,          6 

7,          7 

14,          7 

21,          7 

28,          7 

8,          8 

16,          8 

24,          8 

32,          8 

9,          9 

18,          9 

27,          9 

36>          9 

IO,             IO 

20,         10 

30,        10 

40,         10 

5  is  in 

6  is  in 

7  is  in 

8  is  in 

5,  once. 

6,  once. 

7,  once. 

8,  once. 

IO,     .          2 

12,          2 

14,          2 

16,          2 

15,          3 

18,          3 

21,          3 

24,          3 

20,          4 

24,          4 

28,          4 

32,          4 

25,          5 

30,         5 

35,          5 

40,         5 

30,          6 

36,          6 

42,          6 

48,          6 

35,          7 

42,          7 

49,          7 

56,          7 

40,          8 

48,          8 

56,          8 

64,          8 

45,          9 

54,          9 

63,          9 

72,          9 

50,        10 

60,         10 

70,        10 

80,        10 

9  is  in 

10  is  in 

11  is  in 

12  is  in 

9,  once. 

10,  once. 

11,  once. 

12,  once. 

18,          2 

20,          2 

22,          2 

24,          2 

27,         3 

3°,         3 

33>         3 

36,         3 

36,          4 

40,          4 

44,          4 

48,         4 

45,          5 

5o,          5 

55,          5 

60,         5 

54,          6 

60,          6 

66,          6 

72,          6 

63,          7 

70,          7 

77,          7 

84,          7 

72,          8 

80,          8 

88,          8 

96,          8 

81,          9 

90,          9 

99,          9 

108,          9 

90,        10 

IOO,           10 

no,         10 

120,         10 

*  After  2,  3,  4,  etc.,  in  the  second  column,  "  times "  is  under- 
stood. 


62  DIVISION. 

DEFINITIONS. 

1.  What  is  Division  ? 

Division  is  finding  how  many  times  one  number 
is  contained  in  another. 

2.  What  is  the  number  to  be  divided  called? 

The  Dividend. 

3.  The  number  to  divide  by  ? 

The  Divisor. 

4.  What  is  the  number  obtained  by  division  called  ? 

The  Quotient. 

5.  What  is  the  number  left  called  ? 

The  Remainder* 

When  it  is  said  that  3  is  contained  in  13,  4  times  and 
1  over,  which  is  the  dividend  ?  The  divisor  ?  The  quo- 
tient ?    The  remainder  ? 

Remakks. — 1.  The  remainder  is  always  the  same  denomination 
as  the  dividend;  for,  it  is  a  part  of  the  dividend  not  yet  divided. 
2.  A  proper  remainder  is  always  less  than  the  divisor. 

6.  When  the  dividend  contains  only  one  denomination,  what 
is  the  operation  called  ? 

Simple  Division. 

7.  How  is  Division  denoted? 

By  a  short  horizontal  line  between  two  dots  (—), 
called  the  Sign  of  division. 

8.  When  placed  between  two  numbers  what  does  it  show  ? 

It  shows  that  the  number  before  it  is  to  be  divided  by 
the  one  after  it.  Thus,  21-^-3,  shows  that  21  is  to  be 
divided  by  3,  and  is  read  "21  divided  by  3." 

9.  How  else  is  division  denoted? 

By  writing  the  divisor  under  the  dividend  with  a 
short  line  between  them ;  as  %k 

Read  the  following :  9-7-3  =  3;  24-7-4  =  5  +  1;  39  + 
3  =  "  +  3;  5+4  =  36  +  4;  V  =  7>  ¥  =  4  +  3- 


division.  63 

OBJECTS     OF     DIVISION. 

i.  A  lad  having  6  cents  wishes  to  buy  pears,  which 
are  2  cents  apiece :  how  many  can  he  buy? 

Analysis. — He  can  buy  as  many  pears 
as  there  are  times  2  cents  in  6  cents.    The  Illustbatiow. 

object  then  is  to  find  how  many  times  2     JS   0  I  0  51  I  J3   J3 
is  contained  in  6 ;  and  2  is  in  6,  3  times. 

2.  A  lad  has  6  pears,  which  he  wishes  to  divide  equally 
between  2  companions :  how  many  can  he  give  to  each  ? 

Analysis. — The  object  of  this  example 
is  to  divide  6  pears  into  2   equal  parts.  Illttstbatxon. 

Dividing  6  by  2,  the  quotient  is  3,  which      SJ   21   &  |  &   fil   8 
shows  that  there  are  3  pears  in  each  part. 

10.  What  is  the  object  or  office  of  Division  ? 

Its  object  or  office  is  twofold :  First,  To  find  lioio 
many  times  one  number  is  contained  in  another.  (Ex.  1.) 

Second,  To  divide  a  number  into  equal  parts.   (Ex.  2.) 

Remark. — The  two  preceding  examples  are  representatives  of 
the  two  classes  of  problems  to  which  Division  is  applied.  In  the 
first  class,  the  divisor  and  dividend  are  always  of  the  same  denom- 
ination, and  the  quotient  is  times,  or  an  abstract  number. 

In  the  second,  the  divisor  and  dividend  are  of  different  denom- 
inations, and  the  quotient  is  always  of  the  same  denomination  as 
the  dividend.  This  class  involves  the  idea  of  Fractions,  and  will 
receive  further  attention  under  that  branch  of  the  science. 

Note. — The  process  of  reasoning  in  the  solution  of  these  two 
classes  of  examples  is  somewhat  different ;  but  the  practical  oper- 
t:ion  is  the  same,  viz. :  to  find  how  many  times  one  number  is  con- 
tained in  another,  which  accords  with  the  definition  of  Division. 

11.  How  divide  a  number  into  two,  three,  four,  etc.,  equal  parts  ? 
Divide  the  number  by  2,  3,  4,  5,  etc.,  respectively. 

12.  When  a  thing  is  divided  into  2,  3,  4,  etc.,  equal  parts,  what 
are  the  parts  called  ? 

If  divided  into  two  equal  parts,  the  parts  are  called 
halves  ;  into  three,  the  parts  are  called  thirds  ;  into  four, 
they  are  called  fourths  ;  into  five,  fifths  ;  etc. 


64  division. 

13.  When  a  thiug  is  divided  into  equal  parts,  from  what  do 
the  parts  take  their  name  ? 

From  the  number  of  parts  into  which  the  thing  is 
divided. 

3.  What  is  a  half  of  10  ?  A  third  of  12  ?  A  fourth 
of  16?    A  fifth  of  20?    A  sixth  of  30? 

4.  What  is  a  seventh  of  35  ?  An  eighth  of  56  ?  A 
ninth  of  45  ?    A  tenth  of  60  ?    A  twelfth  of  84  ? 

SHORT      DIVISION. 
MENTAL    EXERCISES. 

1.  How  many  lemons,  at  2  cents  apiece,  can  George 
buy  for  10  cents  ? 

Analysis. — Since  2  cents  will  buy  1  lemon,  10  cents  will  buy 
as  many  lemons  as  2  is  contained  times  in  10 ;  and  2  is  in  10,  5 
times.    Therefore,  he  can  buy  5  lemons. 

2.  At  4  cents  each,  how  many  bananas  can  you  buy 
for  12  cents? 

3.  How  many  yards  of  tape,  at  6  cents  a  yard,  can 
be  had  for  18  cents  ? 

4.  At  4  dollars  a  yard,  how  many  yards  of  cloth  can 
you  buy  for  28  dollars  ? 

5.  When  the  fare  on  the  city  railroads  is  5  cents  a 
ride,  how  many  rides  can  you  take  for  30  cents? 

6.  If  3  oranges  cost  12  cents,  what  will  1  cost? 

Analysis. — If  3  oranges  cost  12  cts.,  1  orange  will  cost  1  third 
of  12  cts. ;  and  1  third  of  12  cts.  is  4  cts.     (P.  63,  Q.  12. 

7.  If  5  slates  cost  60  cents,  what  will  1  cost? 

8.  A  baker  divided  28  loaves  of  bread  equally  among 
7  beggars :  how  many  loaves  did  he  give  to  each  ? 

9.  A  grocer  sold  9  barrels  of  flour  for  72  dollars: 
\t  Lat  waa  thai  a  barrel  ? 


division.  65 

SLATE     EXERCISES. 

When  the  Divisor  is  exactly  contained  in  each  figure  of 
the  Dividend. 

i.  How  many  times  is  2  contained  in  6402  ? 

Analysis. — Write  the  divisor  on  the  left  of  the  Operation. 
dividend,  with  a  curve  line  between  them,  and  pro-  2)6402 

ceed  thus :  2  is  contained  in  6,  3  times ;  write  the  3       .       

under  the  figure  divided,  for  it  is  the  same  order  as  «3 

that  figure.  Next,  2  is  contained  in  4,  2  times ;  write  the  2  under 
the  figure  divided,  for  the  same  reason  as  before.  2  is  contained 
in  o,  no  times ;  write  a  cipher  in  the  quotient.  Finally,  2  is  in  2, 
1  time ;  set  the  1  under  the  figure  divided. 

10.  How  write  numbers  for  division  ? 

Place  the  divisor  on  the  left  of  the  dividend,  with  a 
curve  line  between  them. 

11.  How  proceed  when  the  divisor  is  contained  exactly  in 
each  figure  of  the  dividend  ? 

Begin  at  the  left  of  the  dividend,  and  divide  each 
figure  by  the  divisor  ;  placing  the  result  wider  the  figure 
divided. 

12.  What  order  is  each  quotient  figure  ? 

The  same  order  as  the  figure  divided. 
Copy  and  divide  the  following  in  like  manner : 
M  (3.)  (4.)  (5.) 

3)6393  2)4062  4)8404  5)50505 

(6.)  (7.)  (8.)  (9.)     f 

4)8084  6)6606  7)7070  8)80808 

MENTAL     EXERCISES. 

1.  At  4  dollars  a  head,  how  many  sheep  can  a  man 
buy  with  35  dollars,  and  what  will  he  have  left? 

Analysis. — 4  dollars  are  contained  in  35  dollars  8  times,  and 
3  over.    Therefore,  he  can  buy  8  sheep,  and  have  3  dollars  left. 


66  division. 

2.  How  many  times  is  3  contained  in  1 7,  and  ho\* 
many  over  ? 

3.  In  24  how  many  times  5,  and  how  many  over? 

4.  In  39  how  many  times  4?     5?     6?     7  ?     8?     9? 

5.  How  many  boxes,  each  containing  6  quarts,  can  be 
filled  with  40  quarts  of  blue-berries  ? 

6.  Horace  has  38  marbles,  which  he  wishes  to  distn 
bute  equally  among  his  3  brothers :  how  many  can  he 
give  to  each ;  and  how  many  over  ? 

7.  How  many  times  7  in  29,  and  how  many  over? 

8.  How  many  times  8  in  5  7  ?    In  63  ?    In  74  ?    In  83  ? 

SLATE      EXERCISES. 

When  the  Divisor  is  not  contained  exactly  in  ecch  figure  of 
the  Dividend. 

1.  How  many  barrels  of  flour,  at  5  dollars  a  barrel, 
can  be  bought  for  157034  dollars  ? 

Analysis. — As  the  divisor  is  not  contained  in         Operation. 
the  first  figure  of  the  dividend,  we  m ust  find  how         5 ) x  5  7  °34 

many  times  it  is  contained  in  the  first  two  fig-  

ures,  which  is  3  times,  and  set  the  3  under  the  -A-IIS.  31400^ 
right  hand  figure  divided.  Again,  5  is  contained  in  7,  once  and  2 
remainder.  Set  the  1  under  the  figure  divided,  and  prefixing  the 
2  remainder  mentally  to  the  next  figure  of  the  dividend  makes 
20.  "Now  5  is  in  20,  4  times.  Set  the  4  under  the  figure  divided. 
Next,  5  is  not  contained  in  3,  the  next  figure  of  the  dividend ;  we 
therefore  put  a  cipher  in  the  quotient,  and  prefixing  the  3  men- 
tally to  the  next  figure  of  the  dividend,  makes  34.  Now,  5  is  in 
34,  6  times  and  4  remainder.  We  set  the  6  under  the  figure 
divided,  and  as  there  are  no  more  figures  in  the  dividend,  wc 
write  this  last  remainder  over  the  divisor,  and  annex  it  to  tho 
quotient 

13.  When  the  divisor  is  not  contained  in  the  first  figure  of 
the  dividend,  how  proceed? 

Find  how  many  times  it  is  contained  in  the  first  two 
figures. 


division.  67 

14.  When  it  is  npt  contained  in  a  subsequent  figure  of  the 
dividend,  how  ? 

Put  a  cipher  in  the  quotient,  and  find  how  many 
times  the  divisor  is  contained  in  this  and  the  next  figure. 
setting  the  result  under  the  right  hand  figure  divided. 

3  5.  When  there  is  a  remainder  after  dividing  a  figure,  how  ? 

Prefix  it-  mentally  to  the  next  figure  of  the  dividend, 
and  divide  this  number  as  before.  * 

16.  If  there  is  a  remainder  after  dividing  the  last  figure  of  the 
dividend,  what  is  to  be  done  ? 

Write  it  over  the  divisor,  and  annex  it  to  the  quotient  ? 

Note. — To  prefix  signifies  to  place  before;  to  annex,  to  place 
after. 

1 7.  What  are  the  principles  as  to  the  nature  of  the  divisor  and 
dividend,  the  quotient  and  remainder  ? 

ist.  The  divisor  and  dividend  may  be  abstract 
or  concrete  numbers. 

2d.  When  they  are  the  same  denomination,  the  quo- 
tient denotes  times,  and  is  an  abstract  number. 

3d.  When  they  are  different  denominations,  the  quo- 
tient denotes  equal  parts,  and  is  the  same  denomination 
as  the  dividend. 

4th.  The  remainder  is  always  the  same  denomination 
as  the  dividend ;  for,  it  is  an  undivided  part  of  it. 

18.  What  is  Short  Division? 

Short  Division  is  the  method  of  dividing,  when 
the  results  of  the  several  steps  are  carried  in  the  mind, 
and  the  quotient  only  is  set  down. 

Copy  and  divide  the  following  by  Short  Division  : 
(2)  .     (3-)  (4.)  (50 

4)l2568  3)60429  6)l8728  7)84079 


68  division. 

10.  The  preceding  principles  may  be  summed  up  in 
the  following 

RULE    FOR    SHORT    DIVISION. 

I.  Place  the  divisor  on  the  left  of  the  dividend,  and 
beginning  at  the  left,  divide  each  figure  by  it,  setting  the 
result  under  the  figure  divided, 

11.  If  the  divisor  is  not  contained  in  a  figure  of  the, 
dividend,  put  a  cipher  hi  the  quotient,  and  find  how 
many  times  the  divisor  is  contained  in  this  and  the  next 
figure,  setting  the  result  under  the  right  hand  figure 
divided. 

III.  If  a  remainder  arises  from  any  figure  before  the  last, 
prefix  it  mentally  to  the  next  figure,  and  divide  as  before. 
If  from  the  last,  place  it  over  the  divisor,  and  annex 
it  to  the  quotient 

Pkoof. — Multiply  the  divisor  and  quotient  together, 
and  to  the  product  add  the  remainder.  If  the  result  is 
equal  to  the  dividend,  the  worlc  is  right. 

Note. — This  proof  depends  upon  the  principle,  that  Division  is 
the  reverse  of  Multiplication ;  the  dividend  answering  to  the  pro- 
duct, the  divisor  to  one  of  the  factors,  and  the  quotient  to  the  other. 

EXAMPLES    FOR    PRACTICE. 

i.  At  8  dollars  apiece,  how  many  hats  can  be  bought 
ffor  23243  dollars  ? 

Operation.  Proof. 

8)23243  2905  x  8  =  23240 

~     .  Add  the  remainder,         3 

Quot.  2905,  3  over.  _ 

Ans.    2905  hats,  and  3  dols  over.  Dividend  23243 

(2.)  (3.)  (40  (50 

2)23416  3)34169  4)48016  5)9°3IQ 


DIVISION.  69 

(6.)  (70  (8.)  (9-) 

6)67419  7)75008  8)89619  9)93048 

10.  How  many  barrels  of  apples,  at  3  dollars  a  barrel, 
can  you  buy  for  846  dollars  ? 

11.  At  5  dollars  apiece,  how  many  hats  can  be  bought 
for  2300  dollars  ? 

12.  At  6  dollars  a  barrel,  how  many  barrels  of  flour 
will  3522  dollars  buy? 

13.  At  7  days  each,  how  many  weeks  in  365  days? 

14.  If  a  man  travels  8  miles  per  hour,  how  long  will 
it  take  him  to  travel  1000  miles? 

15.  If  1  boat  will  carry  9  persons  over  a  river,  how 
many  boats  will  it  require  to  carry  468  persons  over? 

16.  If  a  man  lays  up  12  dollars  a  week,  how  long  will 
it  take  him  to  lay  up  288  dollars? 

17.  At  11  dollars  a  barrel,  how  many  barrels  of  cran- 
berries can  be  bought  for  770  dollars? 

18.  How  many  boxes  will  it  require  to  pack  1530 
pounds  of  butter,  allowing  9  pounds  to  a  box  ? 

19.  A  man  left  31265  dollars  to  be  divided  equally 
among  his  5  children :  how  much  did  each  receive  ? 

20.  A  grocer  sold  oranges  at  8  dollars  a  box,  and  re- 
ceived 22464  dollars:  how  many  boxes  did  he  sell? 

21.  If  a  man  has  26436  acres  of  land,  how  many  acres 
can  he  give  to  each  of  his  12  children  ? 

22.  A  company  of  11  men  took  a  prize  worth  1 16633 
dollars,  which  was  equally  divided  among  them :  what 
did  each  receive  ? 

23.  If  a  ship  sails  10  miles  an  hour,  how  many  hours 
will  it  be  in  sailing  25000  miles? 

24.  If  1  stage  will  sea*  12  passengers,  how  many  stages 
will  be  required  to  seat  1500  passengers  ? 


70  DIVISION. 

LONG     DIVISION. 

i.  Divide  22431  by  4,  by  Long  Division. 

Analysis. — Write  the  divisor  on  the  left  Operation. 

of  the  dividend,  as  in  Short  Division,  and    *>>v-  Dividend.      o_uot. 
proceed  thus:   First,  the  divisor  4  is  con-      4)  22431  (  5607  J 
tained  in  22,  5  times  ;  set  the  5  on  the  right  20  ' ' ' 

of  the  dividend  with  a  curve  line  between  — 

them.    Second,  multiply  the  divisor  by  this  24 

quotient  figure,  and  set  the  product  20,  under  24 

the  figure  divided.    Third,  subtract  the  pro-  ' 

duct  from  the  figures  divided,  and  the  re-  ^ 

mainder  is  2.  Fourth,  bring  down  and  annex 
to  the  remainder  the  next  figure  of  the  divi-  ^ 

dend,  making  24  for  the  next  partial  divi- 
dend. Divide  this  partial  dividend,  and  the  quotient  figure  is  b. 
Multiply  and  subtract  as  before,  and  the  remainder  is  o.  Bring 
down  the  next  figure  3  for  a  new  partial  dividend.  But  the  divi- 
sor 4  is  not  contained  in  5  ;  we  therefore  put  a  cipher  in  the  quo- 
tient, and  bringing  down  the  next  figure,  we  have  31  for  a  partial 
dividend,  which  we  divide  as  before.  As  there  are  no  more  figures 
to  be  divided,  we  place  the  last  remainder  over  the  divisor,  and 
annex  it  to  the  quotient.     The  answer  is  5607I. 

Note. — To  prevent  mistakes,  it  is  customary  to  place  a  mark 
under  the  several  figures  of  the  dividend,  when  brought  down. 

20.  What  is  Long  Division  ? 

Long  Division  is  the  method  of  dividing  when 
the  results  of  the  several  steps  and  the  quotient  are  both. 
Bet  down. 

21.  How  write  numbers  for  Long  Division  ? 

Place  the  divisor  on  the  left  of  the  dividend,  and  the 
quotient  on  the  right,  with  a  curve  line  between  them. 

22.  How  many  steps  in  long  division ?    "Four." 
2:5.  The  first? 

Find  hoiv  many  times  the  divisor  is  contained 
in  the  fewest  figures  on  the  left  of  the  dividend  that  will 
contain  it. 


DIVISION.  71 

21.  The  second? 

Multiply  the  divisor  by  the  quotient  figure,  and 
eet  the  product  under  the  figures  divided. 

25.  The.  third? 

Subtract  the  product  from  the  figures  divided. 

26.  The  fourth? 

Annex  to  the  remainder  the  next  figure  of  the  divi 
dend,  for  a  new  partial  dividend ;  then  divide  as  before. 

Remark. — The  quotient  figure  in  Long  and  in  Short  Division, 
is  the  same  order  as  the  right  hand  figure  of  the  partial  dividend. 

2.  Divide  5463  by  4.        Ans.  1365 J. 

3.  Divide  17382  by  5.        4.  Divide  43652  by  6. 

5.  At  45  dollars  an  acre,  how  much  land  can  be 
bought  with  6750  dollars?  Ans.  150  acres. 

6.  How  many  suits  of  clothes,  at  63  dollars  a  suit,  can 
be  had  for  7686  dollars?  Ans.  122  suits. 

27.  The  preceding  principles  may  be  summed  up  in 
the  following 

RULE     FOR    LONG     DIVISION, 

I.  Find  how  many  times  the  divisor  is  contained  in 
the  fewest  figures  on  the  left  of  the  dividend  that  will 
contain  it,  and  set  the  quotient  on  the  right. 

II.  Multiply  the  divisor  by  this  quotient  figure,  and 
subtract  the  product  from  the  figures  divided. 

III.  To  the  right  of  the  remainder,  bring  down  the 
next  figure  of  the  dividend,  and  divide  as  before. 

IV.  If  the  divisor  is  not  contained  in  a  partial  divi- 
dend, place  a  cipher  in  the  quotient,  bring  down  another 
figure,  and  thus  continue  the  operation. 

If  there  is  a  remainder  after  dividing  the  last  figure* 
net  it  over  the  divisor,  and  annex  it  to  the  quotient. 


72  DIVISION". 

Notes. — i.  Long  Division,  is  the  same  in  principle  as  Short. 
The  only  difference  is,  in  one  the  results  of  the  several  steps  are 
carried  in  the  mind,  and  in  the  other  they  are  set  down. 

Short  Division  is  the  more  expeditious,  and  should  be  employed 
when  the  divisor  does  not  exceed  12. 

2.  If  the  product  of  the  divisor  into  the  figure  placed  in  the 
quotient  is  greater  than  the  partial  dividend,  it  is  plain  the  quo- 
tient figure  is  too  large,  and  therefore  must  be  diminlslied. 

3.  If  the  remainder  is  equal  to  or  greater  than  the  divisor,  the 
quotient  figure  is  too  small,  and  must  be  increased. 


EXAMPLES    FOR    PRACTICE. 

1.  How  many  times  is  24  contained  in  1963  ? 

2.  Divide  40369  by  18.  6.  Divide  85345  by  53. 

3.  Divide  45683  by  21.  7.  Divide  906530  by  68. 

4.  Divide  614897  by  35.         8.  Divide  990046  by  74. 

5.  Divide  598061  by  47.         9.  Divide  867604  by  84. 

10.  Eequired  the  quotient  of  9134669  divided  by  92. 

11.  How  many  cows,  at  35   dollars  apiece,  can  be* 
bought  for  7140  dollars? 

12.  How  much  land,  at  28  dollars  per  acre,  can  be 
bought  for  5611  dollars. 

* 3.  If  a  man  earns  45  dollars  a  month,  how  long  will 
it  take  him  to  earn  1620  dollars  ? 

14.  How  many  stoves,  at  38  dollars  each,  can  be  bought 
for  6840  dollars  ? 

15.  If  there  is  1  year  in  52  weeks,  how  many  yean 
are  there  in  8202  weeks? 

16.  If  a  man's  expenses  are  6^  dollars  a  month,  how 
long  can  he  live  on  5260  dollars? 

17.  If  a  man  pay  70  dollars  a  hogshead  for  molasses, 
how  many  hogsheads  can  he  buy  for  6940  dollars  ? 

18.  At  87  dollars  per  yoke,  how  many  yoke  of  qxqu 
can  be  bought  for  6525  dollars? 


DIVISION.  73 

To  find  the  Quotient  Figure,  when  the  Divisor  is  large, 

19.  Divide  12451  by  382. 

Analysis. — Taking  3  for  a  trial  divisor,  it  382  )  1245 1(32 

is  contained  in  12,  4  times.    But  in  multiply-  _  j  .^ 

ing  the  8  by  4,  we  have  3  to  carry,  and  3  added  

to  4  times  3,  make  15,  which  is  larger  than  -    99 1 

the  partial  dividend  12.    Hence,  4  is  too  large  764 

for  the  quotient  figure.     We  therefore  place  

3  in  the  quotient,  and  proceed  as  before.  liem.  227 

2§.  How  find  the  quotient  figure,  when  the  divisor  is  large? 

Take  the  first  figtire  of  the  divisor  for  a  trial  divisor, 
and  find  how  many  times  it  is  contained  in  the  first  or 
first  two  figures  of  the  dividend,  making  dtie  allowance 
for  carrying  the  tens  of  the  product  of  the  second  figure 
of  the  divisor  into  the  quotient  figure, 

20.  Divide  8732409  by  657.       22.  10342675  -f-  3435. 

21.  Divide  9753102  by  950.        23.  23046750-^7625. 

24.  A  certain  fort  has  provisions  sufficient  to  last  1 
man  15360  days  :  how  long  will  it  last  256  men  ? 

25.  The  president's  salary  is  25000  dollars  a  }<ear :  how 
much  is  that  per  day  ? 

26.  At  a  certain  auction,  498  pictures  were  sold  for 
13944  dollars:  what  was  the  average  price  ? 

CONTRACTIONS. 
I.  When  the  Divisor  is  10,  100,  1000,  etc. 

1.  At  100  dollars  a  set,  how  many  sets  of  fur  can  be 
bought  for  1935  dollars,  and  how  much  over  ? 

Analysis. — Annexing  a  cipher  to  Opekation. 
a  number,  multiplies  it  by  10.  (P.  tIoq^  Tn|-C 
56,  Q.  17.)  '      ;    9I35 

Conversely,  removing  a  cipher  or     Quot.  19,  and  35  Rem. 
figure  from  the  right  of  a  number, 

divides  it  by  10;  for,  each  figure  in  the  number  is  removed  one 
place  to  the  right.    (P.  11,  Q.  17.) 
4 


71  DIVISION. 

In  like  manner,  cutting  off  two  figures  from  the  right  of  a  num. 
ber,  divides  it  by  ioo ;  cutting  off  three,  by  iooo,  etc. 

Now  as  the  divisor  is  ioo,  it  is  only  necessary  to  cut  off  two 
figures  on  the  right  of  the  dividend  :  those  left,  viz.,  19,  are  the 
quotient,  and  those  cut  off,  viz.,  35,  the  remainder. 

29.  How  proceed  when  the  divisor  is  10,  100,  1000,  etc.  ? 
From  the  right  of  the  dividend  cut  off  as  many  figures 

as  there  are  ciphers  in  the  divisor.  TJie  figures  left  icid 
he  the  quotient;  those  cut  off,  the  remainder. 

2.  Divide  8564  hyj  100.  6.  39467  by  10000. 

3.  Divide  46531  by  1000.  7.  272364  by  100000. 

4.  Divide  48000  by  1000.  8.  1 000000  by  1 00000. 

5.  Divide  4375681  by  10000.  9.  85325764  by  1000000. 

II.  When  there  ar*e  Ciphers  on  the  right  of  the  Divisor. 

10.  At  20  dollars  apiece,  hovr  many  bureaus  can  be 
bought  for  3453  dollars  ? 

Analysis. — The  divisor,  20,  is  com-  Operation. 

posed  of  the  factors  2  and  10.     In  the       2]°) 345J3 

operation,  we  first  divide  by  io,  by  cut-       A  T"  . 

x.       ^.i     .  v^v    j  *  r*i    a-  •     Ans.  172  b.  13  rem. 

ting  off  the  right-hand  figure  of  the  divi-  '  ° 

dend ;  then  divide  the  remaining  figures  by  2,  the  other  factor 
of  the  divisor.  The  result  172  is  the  quotient;  and  3,  the  figure 
cut  off,  being  annexed  to  the  remainder,  forms  the  true  remainder. 

30.  How  proceed,  when  there  are  ciphers  on  the  right  of  the 
divisor ? 

I.  Cut  off  the  ciphers  on  the  right  of  the  divisor,  and 
as  many  figures  on  the  right  of  the  dividend. 

11.  Divide  the  remaining  part  of  the  dividend  by  the 
remaining  paH  of  the  divisor  for  the  quotient. 

III.  Annex  the  figures  cut  off  to  the  remainder,  and 
(lie  result  will  he  the  true  remainder. 

II.  Divide  8534  by  20.         14.  Divide  23681  by  300. 

12.  Divide  12345  by  30.       15.  Divide  40642  by  130Q. 

13.  163045-^1900.  16.  264168  —  31000. 


division.  75 


DRILL    FOR    RAPID     COMBINATIONS. 

To  Teachers.— These  and  other  drill  exercises,  should  he  continued  hut 
a  few  minutes  at  a  time.  If  spirited  and  frequent,  better  results  will  he 
obtained  from  them,  though  short,  than  from  scores  of  examples  recited  in  an 
indifferent,  sluggish  manner. 

Oral. — i.  To  4  add  8 ;  subtract  2 ;  multiply  by  3 ; 
divide  by  5;  add  4;  multiply  by  3  ;  add  10 ;  result  ? 

2.  From  1 2  subtract  5  ;  add  2  ;  multiply  by  4 ;  divide 
by  6 ;  add  5  ;  multiply  by  3 ;  result  ? 

3.  Multiply  3  by  6 ;  add  4 ;  subtract  2  ;  divide  by  5  ; 
multiply  by  6 ;  add  8  ;  divide  by  4 ;  result  ? 

4.  Divide  42  by  7 ;  multiply  by  4 ;  subtract  6 ;  divide  by 
by  3 ;  add  5  ;  add  9 ;  divide  by  5  ;  multiply  by  1 1 ;  result  ? 

5.  To  14  add  8;  take  4;  divide  by  9;  multiply  by  8; 
add  8 ;  divide  by  8 ;  multiply  by  9 ;  result  ? 

6.  From  27  take  9;  divide  by  9;  multiply  by  9;  add 
9;  take  7 ;  multiply  by  2  ;  divide  by  10;  result  ? 

7.  Multiply  9  by  7;  subtract  7  ;  divide  by  8;  add  12 ; 
subtract  4;  divide  by  5;  multiply  by  12;  divide  by  9; 
add  20 ;  divide  by  6,  multiply  by  1 1 ;  result  ? 

8.  Divide  54  by  9  ;  multiply  by  7  ;  subtract  6 ;  divide 
by  9 ;  multiply  by  8 ;  add  7 ;  subtract  4 ;  divide  by  7  ; 
add  30 ;  result  ? 

9.  Add  7  to  15;  divide  by  11;  multiply  by  9;  add 
10;  divide  by  7;  multiply  by  12;  add  11;  subtract  4; 
divide  by  5  ;  multiply  by  8 ;  result  ? 

10.  Multiply  8  by  7;  subtract  6;  divide  by  10;  mul- 
tiply by  9;  add  n;  divide  by  8;  multiply  by  9;  sub- 
tract 3 ;  add  30 ;  subtract  7  ;  result  ? 

Slate. — 1.  To  36  add  45  ;  subtract  37 ;  multiply  by  6 ; 
divide  by  8 ;  multiply  by  9 ;  add  99 ;  divide  by  9 ;  add 
200 ;  result  ? 

2.  From  87  take  33 ;  multiply  by  7  ;  divide  by  6 ;  add 
233 ;  take  48;  divide  by  8;  multiply  by  25  ;  result? 


76  DIVISION". 

3.  Multiply  348  by  9;  add  556;  divide  by  8;  multi- 
ply Dy  48  j  divide  by  24;  add  545  ;  take  378  ;  result? 

4.  Divide  576  by  24 ;  multiply  by  35  ;  add  1200 ;  di- 
vide by  20 ;  multiply  by  45 ;  divide  by  9 ;  take  375  ; 
add  2375 ;  result? 

5.  To  785  add  357;  take  571;  add  629;  divide  by 
24;  multiply  by  64 ;  divide  by  32  ;  add  873  ;  take  367  ; 
result  ? 

6.  From  3256  take  840  ;  divide  by  302  ;  add  78  ;  mul- 
tiply by  56 ;  divide  by  28 ;  add  1575  ;  result  ? 

7.  Multiply  456  by  28;  divide  by  7;  add  256;  take 
1200;  divide  by  44;  multiply  by  325  ;  result? 

QUESTIONS    FOR     REVIEW. 

Oral. — 1.  If  1  man  can  do  a  job  of  work  in  72  days, 
how  long  will  it  take  9  men  to  do  it? 

Analysis. — 9  men  can  do  9  days  work  in  1  day  :  therefore,  to 
do  72  days  work,  it  will  take  them  as  many  days  as  9  is  con- 
tained times  in  72,  which  is  8.    Ans.  8  days. 

2.  If  a  barrel  of  apples  will  last  1  person  56  days,  bow 
long  will  it  last  a  family  of  7  persons  ? 

3.  How  many  weeks  in  8  times  9  days  ? 

4.  How  many  4-quart  cans  can  be  filled  from  three 
8-quart  pails  ? 

5.  A  farmer  bought  4  pair  of  boots,  at  5  dollars  a  pair, 
and  paid  for  them  in  wheat,  at  2  dollars  a  bushel:  how 
many  bushels  did  the  boots  come  to  ? 

6.  How  many  times  8  in  7  times  12  ? 

7.  A  market-woman  sold  6  dozen  eggs,  at  10  cents  a 
dozen,  and  took  her  pay  in  muslin,  at  12  cents  a  yard: 
how  many  yards  did  she  receive  ? 

8.  How  many  2-gallon  measures  can  be  filled  from  6 
ten -gallon  casks  of  water  ? 


division.  77 

9.  Herbert  bought  20  marbles  at  one  time,  and  16  at 
another;  meantime  he  lost  12 :  how  many  had  he  then  ? 

10.  If  you  earn  9  dollars  a  week,  and  pay  3  dollars  for 
board,  and  2  dollars  for  incidentals,  how  much  will  you 
lay  up  in  9  weeks  ? 

11.  In  40  less  12,  how  many  times  7  ? 

12.  A  trader  bought  12  pair  of  shoes  at  2  dollars,  and 
6  huts  at  5  dollars :  how  much  did  he  pay  for  both  ? 

13.  Three  men  gave  a  poor  person  75  dollars;  one 
gave  30  dollars,  and  another  25  dollars :  how  much  did 
the  other  give  ? 

14.  Three  lads,  counting  their  money,  found  A  had 
25  cents,  B  twice  as  much  as  A,  and  C  as  much  as  both 
the  others :  how  much  had  all  ? 

Slate. — 1.  A  fort  has  provisions  sufficient  to  last  1 
man  365  days:  how  long  will  it  last  a  company  of  73 
men? 

2.  Bought  100  hogs,  weighing  300  pounds  each,  at  7 
cents  a  pound,  and  sold  them  at  10  cents  a  pound:  what 
was  the  profit  ? 

3.  A  grocer  bought  15  hogsheads  of  molasses,  at  35 
dollars  per  hogshead;  151  boxes  of  oranges,  at  6  dollars 
a  box ;  and  91  sacks  of  coffee,  at  20  dollars  a  sack ;  and 
Bold  the  whole  for  4856  dollars:  what  did  he  make  by 
the  operation  ? 

4.  A  farmer  having  115  dollars,  paid  40  dollars  for  a 
cow,  and  the  remainder  for  15  sheep:  what  did  the 
sheep  cost  him  apiece  ? 

5.  A  shoe-dealer  sold  87  pair  of  overshoes,  at  2  dollars 
a  pair;  no  pair  of  boots,  at  9  dollars  a  pair;  and  took 
his  pay  in  coal,  at  n  dollars  a  ton:  how  much  coal 
ought  he  to  receive  ? 

6.  A  teacher  was  engaged  at  1260  dollars  a  year;  at 


78  division. 

the  end  of  9  months  his  health  failed  and  he  left:  how 
much  should  he  receive  ? 

7.  A  grocer  bought  455  barrels  of  flour  for  3185  dol- 
lars; he  afterward  bought  another  lot  at  the  same  rate 
for  1 6 10  dollars:  how  many  barrels  were  there  in  both 
lots;  and  what  did  it  cost  him  per  barrel  ? 

8.  A  man  left  6528  dollars  to  his  wife  and  3  children; 
to  the  latter  he  gave  1265  dollars  apiece:  what  wras  the 
portion  of  his  wife  ? 

9.  A  young  man's  salary  amounted  to  1208  dollars  a 
year  for  3  years;  his  expenses  the  first  year  were  375 
dollars;  the  second,  420  dollars;  and  the  third,  519 
dollars :  how  much  did  he  lay  up  in  the  3  years  ? 

10.  If  I  earn  1350  dollars  a  year,  and  spend  1785  dol- 
lars a  year,  how  much  shall  I  be  in  debt  in  3  years  ? 

1 1.  What  number  taken  from  25973  4- 8230  will  leave 
8768  ? 

12.  What  number  taken  from  41260  —  32S1  will  leave 
8600? 

13.  What  number  taken  from  62135  —  161 2  will  leave 
21500  —  2861  ? 

14.  If  one  man  can  perform  a  piece  of  work  in  750 
days,  how  long  will  it  take  25  men  to  do  it  ? 

15.  If  a  man  earns  1645  dollars  a  year,  and  his  ex- 
penses are  517  dollars  a  year :  how  long  will  it  take  him 
to  lay  up  4512  dollars  ? 

1 6.  Three  lads,  talking  of  their  money,  the  first  said 
he  had  187  cents;  the  second  said  he  had  as  much  as 
the  first  minus  23  cents ;  and  the  third  said  if  he  had 
40  cents  more,  he  should  have  as  many  as  the  other 
two  :  how  many  cents  had  the  second  ?     The  third  ? 

17.  Two  men  being  1950  miles  apart,  traveled  towards 
each  other  at  the  rate  of  35  and  43  miles  a  day  respec- 
tively: how  long  before  they  met? 


FACTOEISG. 

*:!>  Teachers  who  prefer  to  have  pupils  study  United  States  Money  before 
Douimou  and  Decimal  Fractions,  are  referred  to  p.  143. 

i.  What  two  numbers  multiplied  together  make  6? 

2.  What  then  are  the  factors  of  6  ?     (P.  47,  Q.  5.) 

3.  What  are  the  factors  of  10? 

4.  What  are  the  factors  of  8  ?     Of  1 2  ? 

5.  What  are  the  factors  of  14  ?     Of  15  ?     Of  21  ? 

6.  Name  two  factors  of  16.    Of  18.     Of  20. 

7.  Name  two  factors  of  24.     Of  35.     Of  48. 

8.  Name  two  factors  of  54.     Of  63.    Of  72. 

9.  Name  two  factors  of  84.    Of  96.    Of  108. 

DEFINITIONS. 

1.  What  is  a  Factor  ? 

A  Factor  of  a  number  is  one  of  the  numbers,  which 
multiplied  together,  produce  that  number.    (P.  47,  Q.  5.) 

2.  What  is  a  Composite  Number  ? 

A  Composite  Number  is  the  product  of  two  or 
more  factors,  each  of  which  is  greater  than  1.  Thusy 
when  it  is  said  that  3  x  5  =  15,  fifteen  is  a  composite 
number,  and  3  and  5  are  its  factors. 

it.  What  is  a  Prime  Number  ? 

A  Prime  Number  is  one  which  cannot  be  pro- 
duced by  multiplying  any  two  numbers  together,  except 
1  unit  and  itself. 

4.  What  are  Prime  Factors? 

The  Prime  Factors  of  a  number  are  the  prime 
numbers  which,  multiplied  together,  produce  that 
number. 

5.  What  is  an  Odd  Number? 

An  Odd  Number  is-  one  which  cannot  be  divided 
by  2,  without  a  remainder ;  as,  1,  3,  5,  7,  etc. 


80 


FACTORING. 


6.  What  is  an  Even  Number  ? 

An  Even  Number  is  one  which  can  be  divided  by 
2,  without  a  remainder;  as,  2,  4,  6,  8,  etc. 

Note. — All  even  numbers  except  2  are  composite  numbers 

7.  What  is  meant  by  Factoring  a  number  ? 

Factoring  a  Number  is  finding  two  or  more 
factors  which  multiplied  together, produce  that  number 

MENTAL    EXERCISES. 

r.  Name  the  odd  numbers  under  30. 

2.  Name  the  even  numbers  under  30. 

3.  Name  all  the  composite  numbers  under  30. 

4.  Namo  all  the  prime  numbers  under  30. 

5.  What  are  the  prime  factors  of  30  ? 

Analysis. — By  inspection  we  perceive  that  30  is  divisible  by 
the  prime  number  2,  giving  the  factors  2  and  15.  Again,  dividing 
15  by  3  we  have  the  factors  3  and  5,  both  of  which  are  prime 
Therefore,  2,  3,  and  5,  are  the  prime  factors  required. 

6.  What  are  the  prime  factors  of  12?     15?     18? 

7.  What  are  the  prime  factors  of  20  ?     28  ?     30  ? 

8.  What  are  the  prime  factors  of  35  ?  •  40  ?     42  ? 


SLATE     EXERCISES. 
1.  What  are  the  prime  factors  of  105  ? 


OPERATION. 


Analysis. — Dividing  105 
by  the  prime  number  3,  we 
have  the  factors  3  and  35. 
Again,  dividing  the  1st  quo- 
tient 35,  by  the  prime  number 
5,  we  have  the  factors  5  an  J 
7.  Finally,  dividing  the  2d  quotient  7  by  7.  we  have  7  and  1. 
But  the  divisors  3.  5,  and  7  are  all  prime  numbers,  and  therefore 
arc  the  prime  factors  required. 


1st  divisor,  3 
2d  "  5 
3*  "  7 
105  =  3x5*7 


105,  given. 

35,  1st  quot 
7,  2d     " 
i,3d     ■ 


CANCELLATION.  81 

§.  How  is  a  composite  number  resolved  into  prime  factors  ? 

Divide  the  given  number  by  any  prime  number  that 
will  divide  it  without  a  remainder.  Again,  divide  this 
quotient  by  a  prime  number,  and  so  on  till  the  quotient 
is  i.     The  several  divisors  are  the  prime  factors  required. 

Note. — The  least  divisor  of  every  number  is  a  prime  factor; 
hence,  to  avoid  mistakes,  it  is  advisable  for  beginners  to  take  for 
the  divisor,  the  least  number  that  will  divide  the  several  dividends 
without  a  remainder. 

Find  the  prime  factors  of  the  following  numbers : 


2.    42. 

6.  100. 

10.  200. 

14.  625. 

3.    48. 

7.  125. 

11.  256. 

15.  1000. 

4.    60. 

8.  132. 

12.  325. 

16.  1728. 

5-  72. 

9-  175- 

13-  45o- 

— ♦ 

17.  1872. 

CANCELLATION 


i.  What  is  the  quotient  of  3x3x5  divided  by  3  x  5  ? 

Analysis. — 3  X3X  5=45,  and  3X  5  =  15  ;  now  45-^-15=3.  But 
it  will  be  seen  by  inspection  that  the  factors  3  and  5  are  common 
tc  the  dividend  and  the  divisor.  If  we  cancel  or  cross  out  the  3 
in  each,  we  have  3x5-5-5,  or  15-5-5,  which  equals  3,  the  same  as 
before.  Again,  if  we  cancel  or  cross  out  the  5  in  each,  we  have 
3-7-1,  which  equals  3,  as  before. 

Note. — To  cancel  a  factor  of  a  number  means  to  erase  or  reject  it. 

2.  What  is  the  quotient  of  2x3x7-^-2x3x5? 

Solution. — Cancelling  the  common  factors  2  and  3,  we  have 
7-*-5  =  if  A?is. 

1 .  What  is  the  effect  of  cancelling  a  factor  from  a  number  ? 

It  divides  the  number  by  that  factor. 


82  CANCELLATION. 

10.  What  is  Cancellation  ? 

Cancellation  is  the  method  of  abbreviating  opera- 
tions by  rejecting  equal  factors  from  the  divisor  and 
rividend. 

Remakk. — When  the  factor  cancelled  is  equal  to  the  number 
itself,  i  is  always  left  in  its  place ;  for,  dividing  a  number  by  itself, 
the  quotient  is  i.  When  the  i  stands  in  the  dividend,  it  i^ast 
be  retained ;  when  in  the  divisor,  it  may  be  disregarded. 

3.  What  is  the  quotient  of  2x5x7-^2x3  x-7  ? 

Analysis.  —  Writing    the  Operation. 

divisor   under   the  dividend,  1  I 

and  cancelling  the  factors  2  2  X  5  X  ^. I  X  5  X  I  _  s 

and  7,  which  are  common  to  2>  X  3  X  %      1x3x1      ^ 

both,  we  have  1x5x1-^-1x3  j  1 

x  1.    Now  the  product  of  1  x 
5  x  1  =  5,  that  of  1  x  3  x  1=3,  and  5-^-3=1!  Ana. 

11.  What  is  the  rule  for  Cancellation? 

Cancel  all  the  factors  common  to  the  divisor  and  divi* 
de?id,  and  divide  the  product  of  those  remaining  in  tin 
dividend  by  the  product  of  those  remaining  in  the  divisor. 

4.  What  is  the  quotient  of  77  divided  by  21  ? 

Solution. — By  inspection,  we  per-  Operation. 

ceive  that  7  is  a  factor  common  to  the  %%y     j  1 

divisor  and  the  dividend.     Cancelling  ^1     ~~=II~3>0Y  3i 
this  factor,  we  have  V"*  or  3$,  Ans. 

Perform  the  following  divisions  by  cancellation. 

5.  4x5x7-^5x4x3.         8.  28x13x114-11  x  13x7c 

6.  7x3x11-4-8x3x7.       9.  63x39x2-4-13x9x3. 

7.  23x5x9-1-5x7x9.      10.  96x7x11-4-12x8x7. 

1 1.  How  many  yards  of  cloth,  at  8  dollars  a  yard,  can 
be  bought  for  25  pair  of  boots,  at  4  dollars  a  pair  ? 

1 2.  How  many  barrels  of  flour,  at  7  dollars  a  barrel, 
must  be  given  for  18  tons  of  hay,  at  14  dollars  a  ton  ? 

13.  How  long  must  a  man  work,  at  3  dollars  a  day 
to  pay  his  rent  for  a  year,  at  1 1  dollars  a  month  ? 


COMMON    DIVISORS 


MENTAL    EXERCISES. 

i.  What  will  divide  9  and  15  without  a  remainder? 

2.  What  will  divide  14  and  24  without  a  remainder  ? 

3.  What  will  divide  16  and  20  without  a  remainder  ? 

4.  What  will  divide  42  and  18  without  a  remainder  ? 

5.  What  is  the  greatest  divisor  of  18  and  27  ? 

6.  What  is  the  greatest  divisor  of  12  and  36  ? 

DEFINITIONS. 

12.  What  is  a  Common  Divisor? 

A  Common  Divisor  is  a  number  which  will 
Jivide  tivo  or  more  numbers  without  a  remainder. 

13.  What  is  the  Greatest  Common  Divisor? 

The   Greatest  Common  Divisor  of  two  or 

more  numbers,  is  the  greatest  number  that  will  divide 
each  of  them  without  a  remainder. 

Remarks. — 1.  A  common  divisor  of  two  or  more  numbers  is 
always  a  common  factor  of  those  numbers ;  and  the  greatest  com- 
mon divisor  of  them  is  their  greatest  common  factor. 

2.  A  common  divisor  is  often  called  a  common  measure. 

3.  The  greatest  common  divisor  of  two  or  more  numbers  is 
equal  to  the  product  of  all  the  prime  factors  common  to  those 
numbers. 

i.  What  is  the  greatest  common  divisor  of  16  and  20  ? 

1st  Method.— Dividing  the  greater  by  1st  Operation. 
the  less,  the  quotient  is  1,  and  12  remain-  16)28(1 
der.    Again,  dividing  the  first  divisor  by  16 
the  first  remainder  12,  the  quotient  is  1,  T^Wfif  t 
and  4  remainder.     Next,  dividing  the  sec- 
ond divisor  by  the  second  remainder  4,  the  ■ — 
quotient  is  3,  and  o  remainder.     The  last  4  )  T  2  (  3 
divisor  4,  is  the  greatest  common  divisor.  \^_ 


8-1  COMMON     DIVISORS. 

2d  Method. — Setting  the  numbers  in  a  Iwr-  2d  Operation. 

izontal  line,  divide  by  any  prime  number,  as  z,  2)16      28 

that  will  divide  each  of  them  without  a  remain-  ~7~z 

der,  and  set  the  quotients  under  the  correspond-  2  /         T4 

ing  numbers.    Dividing  each  of  these  quotients  4        7 

by  2  again,  the  new  quotients  4  and  7  have  no  Ans.  2x2=4, 
common  factor.     Hence,  the   product  of  the 
common  divisors  2  into  2,  or  4,  is  the  greatest  common  divisor. 

14.  How  find  the  greatest  common  divisor  of  two  or  more 
numbers  ? 

Divide  the  greater  number  by  the  less,  the  first  divisor 
by  the  first  remainder,  the  second  divisor  by  the  second 
remainder,  and  so  on  until  the  remainder  is  nothing ; 
the  last  divisor  will  be  the  greatest  common  divisor. 

Or,  write  the  numbers  in  a  horizorital  line,  and  divide 
by  any  prime  number  that  will  divide  each  without  a 
remainder  ;  setting  the  quotients  in  a  line  below. 

Divide  these  quotients  as  before,  and  thus  proceed,  till 
no  number  can  be  found  that  will  divide  all  the  quotients 
without  a  remainder.  Tfie  product  of  all  the  divisors 
will  be  the  greatest  common  divisor. 

Notes. — 1.  If  there  are  more  than  two  numbers,  and  the  first 
method  is  used,  first  find  the  greatest  common  divizor  of  two  of 
them,  then  of  this  divisor  and  a  third  number,  and  so  on,  until  all 
the  numbers  have  been  used. 

2.  When  there  are  three  or  more  numbers,  the  second  method 
has  the  advantage  both  in  simplicity  and  facility  of  application.  f 

SLATE     EXERCISES. 

2.  Kequired  the  greatest  com.  divisor  of  15, 45,  and  60  ? 

Solution. — Divide  the  given  numbers  by  3)15     45     60 

3,  and  the  quotients  thence  arising  by  5  ;  ~c 

the  next  quotients,  1,  3,  and  4,  have  no  com-         *>  '  *        5 

mon  factor.    Hence,  the  product  of  3  into  5,  134 

or  15,  is  the  answer.  3x5=15,  AtlS. 


COMMON      MULTIPLES.  85 

Find  the  greatest  common  divisor  of  the  following 
numbers : 

2.  27  and  $6.  8.  120  and  148. 

3.  32  and  48.  9.  256  and  512. 

4.  45  and  60.  10.  36,  84,  and  108. 

5.  72  and  24.  11.  45,  60,  and  135. 

6.  75  and  105.  12.  30,  75,  and  225. 

7.  81  and  108.  13.  48,  144,  and  288. 

14.  What  is  the  greatest  number  by  which  128,  160, 
and  192  can  be  exactly  divided  ? 

15.  What  is  the  longest  pole  by  which  108,  132,  and 
144  feet  can  be  exactly  measured? 

16.  A  shopkeeper  has  three  balls  of  twine,  containing 
120,  100,  and  200  yards,  which  he  wishes  to  cut  into 
kite-lines  of  equal  length :  what  is  the  greatest  length 
he  can  make  them  9 


COMMON    MULTIPLES. 


MENTAL    EXERCISES. 

1.  WHat  numbers  under  12  can  be  divided  by  2  with- 
out a  remainder  ? 

2.  By  what  numbers  can  12  be  exactly  divided? 

3.  What  numbers  under  20  can  be  exactly  divided 
by  4? 

4.  By  what  numbers  can  15  be  exactly  divided  ? 

5.  By  how  many  numbers  can  18  be  exactly  divided  ? 

6.  By  what  numbers  can  24  be  exactly  divided  ? 

7.  By  what  two  factors  can  33  be  exactly  divided  ? 

8.  By  what  two  factors  can  35  be  exactly  divided  ? 

9.  What  4  numbers  will  exactly  divide  42  ? 

10.  Name  two  numbers  that  can  be  exactly  divided 
by  4,  5,  and  6. 


Ob  COMMON      MULTIPLES. 

DEFINITIONS. 

15.  What  is  a  Multiple  ? 

A  Multiple  is  a  number  which  can  be  divided  by 
another  number  without  a  remainder. 

1 6.  What  is  a  Common  Multiple  ? 

A  Common  Multiple  is  a  number  which  can  be 
divided  by  two  or  more  numbers  without  a  remainder. 
Thus,  15  is  a  common  multiple  of  3  and  5. 

Remark. — A  common  multiple  of  two  or  more  numbers,  con- 
tains all  the  prime  factors  of  those  numbers. 

17.  What  is  the  Least  Com.  Multiple  of  two  or  more  numbers  ? 

The  Least  Common  Multiple  of  two  or  more 
numbers,  is  the  least  number  which  can  be  divided  by 
each  of  them  without  a  remainder.  Thus,  12  is  the 
least  common  multiple  of  2,  3,  and  4. 

1.  What  is  the  least  com.  multiple  of  12,  18,  and  21  ? 

1ST  Method. — Writing  the  numbers  1st  Operation. 

in  a  horizontal  line,  we  divide  by  any  3)12      18     21 

prime  number  3,  which  will  divide  two 

or  more  of  them  without  a  remain-  2 )  4       6       7 

der,  and  set  the  quotients  in  the  line  "^       I     ~~Z 

below.    Again,  dividing  these  quotients  _ 

by  the  prime  number  2,  which  will  di-    ^  .57—5 

vide  two  of  them  without  a  remainder,  we  set  the  quotients  and 
undivided  number  7  in  a  line  below  as  before.  As  the  numbers 
2,  3,  and  7,  are  prime  factors,  the  division  can  be  carried  no  fur- 
ther. Finally,  the  continued  product  of  the  divisors  and  numbers 
in  the  last  line,  3x2x2x3*  7=252,  is  the  least  com  multiple. 

2d  Method.  .—  Resolve   the    given  2d  Operation. 

numbers  into  their  prime  factors,  as  in  12  =  2x2x3 

the  margin.    But  we  have  seen  that  a  l8=2x$X3 

common  multiple  of  two  or  more  num-  2 1  =$  X  7 

bers  contains  all  the  prime  factors  of  2X2X-X-X,_ 2  c  2 
those  numbers.    Hence,  it  must  contain 

the  prime  factors  of  12.  which  are  2x2x3;  we  therefore  retain 

these  factors.    Again,  it  must  contain  the  prime  factors  of  18, 


COMMON     MULTIPLES.  87 

which  are  2x3x3.  But  we  already  have  two  2s  and  one  3 ;  we 
may  therefore  cancel  the  2,  and  one  of  the  3s,  retaining  the  other 
3.  Finally,  it  must  contain  the  factors  of  21,  which  are  3x7. 
But  since  we  have  retained  two  3s,  we  may  cancel  this  3,  and  re- 
tain the  7.  The  continued  product  of  the  uncancelled  factors 
2X2X3X3X  7=252,  the  same  as  before. 

1§.  How  find  the  least  common  multiple  of  two  or  mon 
numbers  ? 

Write  the  numbers  in  a  horizontal  line,  and  divide  by 
any  prime  number  that  ivill  divide  tiuo  or  more  of  them 
without  a  remainder,  placing  the  quotients  and  numbers 
undivided  in  a  line  below. 

Next  divide  this  line  as  before,  and  thus  proceed  till  no 
two  numbers  are  divisible  by  any  number  greater  than  1. 
Tlie  continued  product  of  the  divisors  and  numbers  in  the 
last  line  ivill  be  the  answer. 

Or,  resolve  the  given  numbers  into  their  prime  factors  ; 
multiply  these  factors  together,  talcing  each  the  greatest 
number  of  times  it  occurs  in  either  of  the  given  numbers, 
and  the  product  ivill  be  the  answer. 

Remark. — Both  of  these  methods  are  based  upon  the  principle, 
that  the  least  common  multiple  of  two  or  more  numbers  is  the 
least  number  which  contains  all  their  prime  factors,  each  factor 
being  taken  as  many  times  as  it  occurs  in  either  of  the  given  num- 
bers. 


SLATE  EXERCISES. 
Find  the  least  common  multiple  of  the  following 

2.  4,  8,  12.  8.  39,  52,  13. 

3.  l6,   12,  24.  9.    8l,   I08,  72. 

4-  i5>  3°>  45-  IO-  24,  12,  48,  60. 

5.  36,  48,  84.  11.  14,  42,  28,  56. 

6.  40,  45>  75-  I2-  54,  81,  96,  120. 

7.  20,  60,  55.  13.  72,  144,  288,  432. 


FRACTIONS 


INTRODUCTORY     EXERCISES. 

To  Teachers.— The  object  Of  this  Exercise  is  to  develop  the  idea  of 
Fractional  parts.  The  best  way  to  secure  this  end  is,  to  let  beginners  divide 
some  object,  as  a  sheet  of  paper,  or  an  apple,  into  halves,  thirds,  fourths, 
etc. ;  then  put  the  parts  together  and  form  the  whole  again. 

i.  If  you  divide  a  sheet  of  paper  into  tivo  equal  parts, 
what  is  each  part  called  ? 
One  half. 

2.  Draw  a  line  an  inch  long  upon  your  slate  or  black- 
board, and  divide  it  into  halves. 

3.  If  you  divide  an  apple  into  three  equal  parts,  what 
is  one  of  the  parts  called  ? 

One  third. 

4.  Two  of  the  parts  ? 
Two  thirds. 

5.  Into  how  many  halves  can  you  divide  an  apple? 
Into  how  many  thirds  ? 

6.  Draw  a  line  a  foot  long,  and  divide  it  into  halves. 
Into  thirds. 

7.  How  many  thirds  make  a  whole  one  ? 

8.  If  a  sheet  of  paper  is  divided  into  four  equal  parts, 
what  is  one  of  the  parts  called  ? 

A  Fourth,  or  quarter. 
\    9.  What  are  two  of  the  parts  called  ?    Three  of  the 
♦parts  ?    How  many  fourths  in  a  whole  sheet  ? 
1    10.  When  a  thing  is  divided  in  five  equal  parts,  what 
are  the  parts  called  ? 

Fifths. 

11.  When  a  thing  is  divided  in  six  equal  parts,  what 
are  the  parts  called  ?  If  divided  in  seven,  what  ?  Into 
eight,  what  ?  Into  nine,  what  ?  Into  ten,  what  ?  Into 
twenty,  what  ?     Into  fifty,  what  ? 


FRACTIONS.  89 

12.  Which  are  greater,  halves  or  thirds  ?  Thirds  or 
fourths?     Sixths  or  fifths  ?     Tenths  or  eighths  ? 

DEFINITIONS. 

1.  What  is  an  Integer  ? 

An  Integer  is  a  number  which  contains  one  or  more 
entire  units  only;  as  i,  3,  5,  8,  12,  etc. 

2.  What  is  a  Fraction  ? 

A  Fraction  is  one  or  more  of  the  equal  parts  into 
which  a  unit  is  divided. 

3.  What  is  meant  by  one-half? 

One  of  the  two  equal  parts  into  which  a  unit  is 
divided. 

What  is  meant  by  a  third  ?  Two  thirds  ?  A  fourth  ? 
Fifth  ?    Seventh  ?    Tenth  ? 

4.  From  what  do  these  parts  take  their  name  ? 

The  Number  of  equal  parts  into  which  the 
unit  or  thing  is  divided. 

5.  Upon  what  does  their  value  depend  ? 

First.  Upon  the  magnitude  of  the  unit  or  tldng 
divided. 

Second.  Upon  the  number  of  parts  into  which 
it  is  divided. 

Illustrate  these  two  points : 

1st.  If  a  large  and  a  small  sheet  of  paper  are  each  divided  into 
halves,  thirds,  fourths,  etc.,  it  is  plain  that  the  parts  of  the  former 
will  be  larger  than  the  corresponding  parts  of  the  latter. 

2d.  If  one  of  two  equal  sheets  of  paper  is  divided  into  two  equal 
parts,  and  the  other  into  four,  the  parts  of  the  first  will  be  twice 
as  large  as  those  of  the  second ;  if  one  is  divided  into  two  equal 
parts,  the  other  into  six,  one  part  of  the  first  will  be  equal  to 
three  of  the  second,  etc.    Hence, 

Note. — 1.  A  half  is  twice  as  large  as  a  fourth,  three  times  as  large 
as  a  sixth,  four  times  as  large  as  an  eighth,  etc. ;  and  generally, 


90  FRACTIONS. 

2.  The  greater  the  number  of  equal  parts  into  which  the  unit 
is  divided,  the  less  will  be  the  value  of  each  part.     Conversely, 

3.  The  less  the  number  of  equal  parts,  the  greater  will  be  the 
value  of  each  part. 

FINDING    FRACTIONAL    PARTS. 

1.  What  is  1  half  of  10  dollars? 

Analysis. — If  10  dollars  are  divided  into  two  equal  parts,  one  o. 
these  parts  is  5  dollars.    Therefore,  &c.    (P.  63,  Q.  II.) 

2.  What  is  1  half  of  8  peaches  ? 

3.  What  is  a  third  of  9  ?    Of  15  ?    Of  1 8  ?    Of  24  ? 

4.  What  is  a  fourth  of  12  ?     Of  16  ?    Of  28  ?    Of  40  ? 

5.  What  is  a  fifth  of  20  ?    Of  45  ?    Of  35  ?    Of  60  ? 

6.  What  is  an  eighth  of  32  ?   Of  56  ?    Of  64?   Of  72  ? 

7.  What  is  2  thirds  of  18  yards  ? 

Analysis. — 2  thirds  are  twice  1  third ;  now  1  third  of  18  yards 
is  6  yards,  and  2  times  6  yards  are  12  yards.     Therefore,  etc. 

8.  What  is  3  fourths  of  32  ? 

9.  What  is  3  eighths  of  48  ? 

10.  What  is  4  fifths  of  35  ? 

11.  What  is  7  tenths  of  60  ? 


NOTATION    OF     FRACTIONS, 

6.  Into  what  two  classes  are  fractions  divided  ? 

Into  Common  and  Decimal. 

7.  What  is  a  Common  Fraction? 

A  Common  Fraction  is  one  in  which  the  unit 
is  divided  into  any  number  of  equal  parts. 

8.  How  are  common  fractions  usually  expressed  ? 

By  Figures  written  above  and  below  a  line,  called 
the  numerator  and  denominator;  as,  f ,  J,  -fa. 


FRACTIONS.  91 

9.  Where  is  the  Denominator  placed,  and  what  does  it  show  ? 
The  Denominator  is  written  lelow  the  line,  and 

shows  into  liow  many  equal  parts  the  unit  is  divided. 

10.  Where  the  Numerator,  and  what  does  it  show  ? 

The  Numerator  is  written  above  the  line,  and 
shows  how  many  parts  are  expressed  by  the  fraction. 

Notes.—  i.  The  denominator  is  so  called  because  it  names  the 
parts ;  as,  halves,  thirds,  etc. 

2.  The  numerator  is  so  called  because  it  numbers  the  parts 
taken.  Thus,  in  the  fraction  §,  3  is  the  denominator,  and  shows 
that  the  unit  is  divided  into  3  equal  parts ;  2  is  the  numerator, 
and  shows  that  2  of  the  parts  are  taken. 

11.  What  are  the  Terms  of  a  fraction? 

The  Terms  of  a  Fraction  are  the  Numerator  and 
Denominator. 

WRITTEN     EXERCISES. 

1.  How  express  one-half,  one-third,  three-fourths,  etc., 
by  figures  ?    Ans.  \,  ±  f . 

Write  the  following  fractions  in  figures : 

2.  Two  thirds.  9.  Ten  twelfths. 

3.  Four  fifths.  10.  Eight  twenty- thirds. 

4.  Three  sevenths.  11.  Mne  thirty-firsts. 

5.  Seven  eighths.  12.  Twenty-three  fortieths. 

6.  Five  ninths.  13.  Nineteen  seventy-fifths. 

7.  Seven  thirds.  14.  Seventy-four  hundredths. 

8.  Four  fourths.  15.  Ninety-nine  thousandths. 

Copy  and  read  the  following : 

16."  -J.  ii.Jf.  26.  |J,  31.  i3|, 

17.  tV       22.  j}.       27.  **        32.  m- 

*«•  A-  23.  if.  28. -v-  33.  «*. 


19.  ^.  24.  «.  29.  -v-.  34.  iff 

20.  ^  25.   Jf  30.  ^.  35,  .£00.. 


9-  FRACTIONS. 

DEFINITIONS. 

12.  Into  what  are  common  fractions  divided? 

Into  proper,  improper,  simple,  compound,  complex  frac- 
tions, and  mixed  numbers, 

13.  Explain  each. 

A  Proper  Ft*action  is  one  whose  numerator  is 
less  than  the  denominator ;  as,  |,  j. 

An  Improper  Fraction  is  one  whose  numerator 
equals  or  exceeds  the  denominator ;  as,  f ,  § . 

A  Simple  Fraction  is  one  having  but  one  numer- 
ator and  one  denominator,  each  of  which  is  a  whole 
number,  and  may  be  propter  or  improper  ;  as,  J,  £. 

A  Compound  Fraction  is  a  fraction  of  a  frac- 
tion ;  as  \  of  \. 

A  Complex  Fraction  is  one  which  has  a  frac- 
tional numerator,  and  an  integral  denominator:  as, 
|  £l* 
3'   A 

A  Mixed  Number  is  a  whole  number  and  a  frac- 
tion expressed  together;  as,  5$,  342]. 

1.  What  kind  of  fractions  are  f ,  $ ,  and  J  ?    Why  ? 

2.  What  kind  are  f  and  J  ?    j  and  |  ?    Why  ? 

3.  What  kind  are  J  of  J  ?    f  of  f  ?    Why  ? 

4.  What  do  you  call  4h  ih  9s  ?    Why  ? 

5.  What  do  you  call  J%  t  ?    Why  ? 

4    5 

11.  What  is  the  value  of  a  fraction  ? 

The  Value  of  a  Fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator.  Thus,  the 
value  of  1  half  is  i-i-2;  of  2  thirds,  is  2-7-3;  °f  4 
fourths,  is  4-1-4,  or  1 ;  of  6  thirds,  is  6^-3,  or  2,  etc. 

*  See  New  Practical  Arithmetic,  Rem.  p.  101. 


FRACTIONS.  93 

GENERAL    PRINCIPLES    OF    FRACTIONS. 

1 5.  What  are  some  of  the  principles  upon  which  the  opera- 
tions in  fractions  depend  ? 

1.  Multiplying  the  numerator  by  any  number,  multi- 
plies the  fraction  by  that  number. 

II.  Dividing  the  numerator,  divides  the  fraction. 

III.  Multiplying  the  denominator,  divides  the  frac- 
tion. 

IV.  Dividing  the  denominator,  multiplies  the  fraction. 

V.  Multiplying  or  dividing  both  the  numerator  and 
denominator  by  the  same  number,  does  not  alter  the  value 
of  the  fraction. 

REDUCTION     OF     FRACTIONS. 

MENTAL    EXERCISES. 

i.  If  I  divide  an  apple  into  halves,  how  can  I  express 
one  of  these  parts  by  figures  ? 

By  \.    (The  pupil  writes  it  upon  the  blackboard.) 

2.  If  you  multiply  both  terms  of  \  by  2,  what  will  it 
become  ? 

It  will  become  j. 

3.  If  you  multiply  both  terms  of  \  by  3, 4. 5,  etc.,  what  ? 
It  will  become  f ,  f ,  T5^,  -^,  and  so  on. 

4.  How  many  fourths  in  \  ? 
Two  fourths. 

5.  How  many  sixths  in  \  ?  How  many  eighths  ? 
How  many  tenths  ?    Twelfths  ? 

6.  If  you  multiply  both  terms  of  \  by  2,  what  will  it 
become  ? 

It  will  become  f . 

7.  If  you  multiply  both  terms  of  \  by  3,  4,  5,  etc., 
what  will  it  become  ? 

It  will  become  % ,  ^,  *fc,  etc. 


94  BEDUCTIONOF 

8.  How  many  thirds  in  }  ?    In  ■&  ?    T6?  ?    -J %  ? 

9.  If  I  divide  an  orange  into  4  equal  parts,  how  can  I 
express  one-half  of  these  parts  by  figures  ? 

By  J.     (The  pupil  writes  it  upon  the  blackboard.) 

10.  If  you  divide  both  terms  of  j  by  2,  what  will  it 
become  ? 

It  will  become  £. 

11.  To  how  many  halves  are  f  equal  ?    -^9    -fa  ? 

1 2.  To  how  many  thirds  are  £  equal  ?    f  ?    T8j  ? 

13.  How  many  fourths  equal  £  ?    T6^  ?    -J^  ? 

DEFINITIONS. 

16.  TOdtf  is  Reduction  of  Fractions? 

Reduction  of  Fractions  is  changing  the  terms, 
without  altering  the  value  of  the  fractions. 

17.  What  is  meant  by  reducing  a  fraction  to  higher  terms? 

It  is  changing  its  numerator  and  denominator  to 
larger  numbers,  without  altering  its  value, 

18.  What  by  reducing  a  fraction  to  loxcer  terms  ? 

It  is  changing  its  numerator  and  denominator  to 
smaller  numbers,  without  altering  its  value. 

19.  What  is  the  principle  upon  which  these  changes  are  made  ? 

The  principle  that  multiplying  or  dividing  both 
he  numerator  and  denominator  by  the  same  number 
ioes  not  alter  the  value  of  the  fraction.     (P.  93,  Pr.  V.) 

CASE    L* 
To  reduce  a  Fraction  to  higher  terms. 

1.  Keduce  f  to  twentieths. 

Analysis. — The  required  denominator  20,  Operation. 

contains  5,  the  given  denominator,  4  times.  20-^5=4. 

But  if  both  terms  of  a  fraction  are  multiplied     3x4 
by  the  same  number,  its  value  is  not  altered.     -  X4  =  a&  AnS* 
Therefore,  multiplying  both  terms  of  \  by  4, 
we  have  %=$%,  the  fraction  required.    (P.  93,  Prin.  V.) 


FRACTIONS.  95 

20.  How  reduce  a  fraction  to  higher  terms  ? 

Multiply  both  terms  of  the  fraction  by  such  a  number 
as  will  make  the  given  denominator  equal  to  the  required 
denominator. 

Note. — The  multiplier  is  found  by  dividing  the  proposed  de- 
nominator by  the  given  denominator. 

2.  Eeduce  f  to  thirtieths. 

3.  Reduce  £  to  fortieths. 

4.  Reduce  |  to  sixty-thirds; 

5.  Reduce  ^  to  fifty-fifths. 

6.  Reduce  ^  to  seventy-fifths. 

7.  Reduce  ■£?  to  seyenty-sixths. 

8.  Reduce  %}  to  one-hundred-and- thirty-fifths. 

9.  Reduce  ff  to  two-hundred-and-sixty-eighths. 
10.  Reduce  -^T  to  one-thousandths. 

CASE    II. 
To  reduce  a  Fraction  to  lower  terms. 

11.  To  how  many  tenths  are  |£  equal  ? 
Analysis. — The   given  denominator  20  Operation. 

contains   10,   the  required  denominator,   2  20-j-lo=2. 

times.     But,  if  both  terms  of  a  fraction  are     12-^2 

divided  by  the  same  number,  its  value  is  _;_    —  TU>  Ans. 

not  altered ;  therefore,  dividing  both  terms 

of  \%  by  2,  it  becomes  -,%,  the  fraction  required.   (P.  93,  Prin.  V.) 

21 .  How  is  a  fraction  reduced  to  lower  terms  ? 
Divide  both  terms  by  such  a  number  as  will  make  the 
yiven  denominator  equal  to  the  required  denominator. 

12.  Reduce  ^  to  fourths.  17.  Reduce  37T  to  fifths. 

13.  Reduce  ^  to  thirds.  18.  Reduce  Jf  to  ninths. 

14.  Reduce  ■£$  to  eighths.  19.  Reduce  ff  to  sixths. 

15.  Reduce  ■£$  to  sixths.  20.  Reduce  -£f  to  eighths. 

16.  Reduce  Jf  to  fourths.  21.  Reduce  T8o\  to  twelfths. 


96  REDUCTION     OF 

CASE    III. 
To  reduce  a  Fraction  to  its  lowest  term9. 

22.  What  are  the  lowest  terms  of  a  fraction  ? 

The  Lowest  Terms  of  a  fraction  are  the  smallest 
numbers  in  which  its  numerator  and  denominator  can 
be  expressed. 

i.  What  are  the  lowest  terms  to  which  -ff  can  be 
reduced  ? 

Analysis. — Dividing  both  terms  of  a  frac-      1st  Operation. 
tion  by  the  same  number,  does  not  alter  its     2  )-J-|-=-|  and 
value.     (Prin.   V.)     Now  if   we  divide    both     2  \  6  —  3    A?IS. 
terms  of  \%  by  2,  we  have  $ .    Again,  dividing 
both  terms  of  the  new  fraction  f  by  2,  the  result  is  f ,  which  are 
the  lowest  terms  in  which  ||  can  be  expressed. 

Or,  we  may  divide  both  terms  of  the  frac-        2d  Operation. 
tion  by  their  greatest  common  divisor,  which    4)^2=^  Ans. 
is  4,  and  obtain  the  same  result.    (P.  84,  Q.  14.) 

23.  How  reduce  a  fraction  to  its  lowest  terms  ? 

Divide  the  numerator  and  denominator  continually  by 
any  number  that  tvill  divide  both  tvithout  a  remainder, 
until  no  number  greater  than  1  will  divide  them. 

Or>  divide  both  terms  of  the  fraction  by  their  greatest 
common  divisor.    (P.  84,  Q.  14.) 

2.  What  are  the  lowest  terms  to  which  £§  can  be  re- 
duced  ?    Ans.  f . 

Keduce  the  following  fractions  to  their  lowest  terms  : 

3.  IS- 

4.  ft- 

5-  it- 

6-  » 
7.  » 
&  if 


n      42 
9-    T2' 

«s» 

21.  -,?8v. 

10.  «. 

16.  T%. 

».  m- 

11.  If- 

■7-  m- 

23.  H* 

12-  Tin- 

18.  T<&. 

24.  «f 

13.  AV 

19-  j*. 

*s-m 

•4-  tt* 

29.  T^. 

»«■  A'iV 

FRACTIONS.  97 

CASE     IV. 

To  reduce  an  Improper  Fraction  to  a  Whole  or  Mixed 

Number. 

i.  In  7  half  dimes,  how  many  whole  dimes  ? 

Analysis. — Since  in  2  halves  there  is  1  whole  dime,  in  7  halves 
there  are  as  many  dimes  as  2  is  contained  times  in  7,  which  is  3 
and  1  half  over,  or  3^  times.  Therefore,  in  7  half  dimes  there  are 
3$  dimes. 

2.  In  10  half  dollars,  how  many  dollars  ? 

3.  To  what  whole  number  is  ty  equal  ? 

4.  To  what  mixed  number  is  -^  equal  ? 

5.  Seduce  ^f-  to  a  whole  or  mixed  number. 

6.  Eeduce  Afl  to  a  whole  or  mixed  number. 

SLATE    EXERCISES. 

1.  Reduce  -^  to  a  whole  or  mixed  number. 
Analysts. — Since  in  3  thirds  there  is  a  unit,  or  one,      operation. 

in  25  thirds  there  are  as  many  units  as  3  is  contained  3  )  2  5 

times  in  25.     Dividing  the  numerator  25  by  the  de-  7 

nominator  3,  the  quotient  is  8  and  1  over,  or  8£.  Alls.  o-£ 
Therefore,  %5  =  8$. 

21.  How  reduce  an  improper  fraction  to  a  whole  or  mixed 
number  1 

Divide  the  numerator  by  the  denominator,  and  the 
quotient  to-ill  be  the  number  required. 

Reduce  the  following  to  whole  or  mixed  numbers  : 

2.  4f.  6.  ¥•  i°-  W«  H.  -¥i2-- 

3.  ¥•  7.  ***•  »-W*  J5-  W- 

4.  *£.  8.  -W-  12.  ^.  16.  W.O, 

5-  ¥•  9.  -W  *3-  -5#-  i7-  ^c?- 

i3.  How  many  sheets  of  paper  shall  I  require,  to  give 

5  pupils  half  a  sheet  apiece  ? 

19.  A  man  meeting  15  beggars,  gave  each  a  quarter 

of  a  dollar :  how  many  dollars  did  he  give  to  all  ? 
5 


98  REDUCTION     OF 

CASE    V. 

To  reduce  a  Whole  or  Mixed  Number  to  an  Improper 
Fraction. 

i.  How  many  halves  in  2  pears  ? 

Analysis. — In  1  pear  there  are  2  halves ;  therefore,  in  2  pears 
there  are  2  times  2  halves,  which  are  4  halves. 

2.  How  many  halves  in  3  whole  ones  ?    In  4  ?    In  5  ? 

3.  In  5  how  many  thirds  ?    In  6  ?    In  7  ?    In  10  ? 

4.  How  many  fourths  in  5  ?    In  7  ?    In  1 1  ?    In  12  ? 

5.  How  mnny  fifths  in  8  ?     In  9  ?    In  12  ? 

6.  How  many  tenths  in  7  ?     In  9  ?    In  10  ? 

7.  How  many  thirds  in  4!  ? 

Analysts. — Since  in  1  there  are  3  thirds,  in  4  there  must  be 
3  times  4,  or  12  thirds,  and  2  thirds  are  14  thirds.  Therefore,  in 
4!  there  are  LaA. 

8.  How  many  fourths  in  5 J  ?    In  6J  ? 

9.  How  many  sevenths  in  5$ ?    In  6§  ?    In  8^? 

10.  Eeduce  8 J  to  an  improper  fraction. 

11.  Reduce  iof  to  an  improper  fraction. 

12.  Reduce  i2f  to  an  improper  fraction. 


SLATE     EXERCISES.  » 

i.  Reduce  17  to  fifths. 

Analysis. — Since  there  are  5  fifths  in  a  unit,  Operation. 
there  must  be  5  times  as  many  fifths  in  a  num-      17  X  5  —  &5* 

ber  as  there  are  units  in  that  number;  and  5  Ans.  ^5. 
times  17  are  85.     Therefore,  17= V» 

2.  Reduce  13^  to  an  improper  fraction ;  that  is,  to 
fifths. 

Analysis. — Reasoning  as  before,  in  13  units  thoro  133. 

are  5  times  13,  or  65  fifths,  and  4  fifths  are  69  fifths.  ,- 

We  therefore  multiply  the  whole  Dumber  13,  by  the  

denominator  5,  and  adding  the  4  fifths  to  the  product,  69 

wo  have  ^.  Ans.  -V- 


FRACTIONS.  99 

25.  How  reduce  a  whole  or  mixed  number  to  an  improper 
fraction  ? 

Multiply  the  tvhole  number  by  the  given  denominator  ; 
to  the  product  add  the  numerator,  and  place  the  sum  over 
the  denominator. 

Note. — A  whole  number  may  be  reduced  to  an  improper  frac- 
tion, by  making  i  its  denominator.  Thus,  3=f;  for  multiplying 
or  dividing  a  number  by  i,  does  not  alter  its  value. 

3.  Eeduce  25  to  thirds.      5.  Seduce  31  to  fourths. 

4.  Eeduce  43  to  fifths.        6.  Eeduce  65  to  sevenths. 

Eeduce  the  following  to  improper  fractions : 


7-  iif- 

12. 

5<5f 

17. 

100-2V 

22. 

iooof. 

8.  i4f. 

13. 

m 

18. 

1 17  -h- 

23- 

I064I. 

9.  i7l- 

14. 

89fV 

19, 

245iV- 

24. 

2207J. 

10.  3 Ji- 

15. 

73tV 

20. 

430yV 

25- 

3°46|. 

ii.  44-£- 

16. 

97tt- 

21. 

5°7tw 

26. 

523*tV 

27.  A  gentleman  having  25  J  dollars,  divided  it  equally 
among  a  company  of  beggars,  giving  each  1  fourth  of  a 
dollar :  how  many  were  there  in  the  company  ? 

CASE    VI. 
To  reduce  a  Compound  Fraction  to  a  Simple  one. 

1.  To  what  is  §  of  -}  equal  ? 

Analysis. — ^  of  £  is  equal  to  -,'5- ;  for  multiply-        Operation. 
ing  the  denominator  divides  the  fraction.    Now,      2  0f  I—  2 
if  ^  of  £  is  iV,  2  thirds  will  be  twice  as  much,  and 
2  times  iL5  are  -,%>.    In  the  operation,  we  multiply  the  numerators 
together  for  the  new  numerator,  and  the  denominators  together  for 
the  new  denominator. 

2.  Eeduce  |  of  J  to  a  simple  fraction? 
Solution. — f  x  i=2%,  or  -,'„,  Ans. 

3.  Eeduce  J  of  }  of  f  to  a  simple  fraction. 


100  DEDUCTION     OF 

4.  Reduce  J  of  £  of  -J  to  a  simple  fraction. 

Analysis. — We  have  seen  that  Operation. 

dividing  the  numerator  and  de-  I      S     4 

nominator  by  the  same  number  y  0I  ?  01  's^=^XilXrZ^, 
ioes  not  alter  the  value  of  the 

fraction ;  also  that  cancelling  a  factor  divides  a  number  by  that 
factor.  We  therefore  cancel  the  common  factors  3  and  4,  then 
multiplying  the  factors  remaining  in  the  numerators  together  for 
the  new  numerator,  and  those  remaining  in  the  denominators  for 
the  new  denominator,  we  have  £  for  the  simple  fraction  required. 

26.  How  reduce  a  compound  fraction  to  a  simple  one  ? 

Cancel  the  common  factors,  and  place  the  product  of 
the  factors  remaining  in  the  numerators  over  the  product 
of  those  remaining  in  the  denominators. 

Note. — The  object  of  cancelling  the  common  factors  is  two- 
fold ;  it  shortens  the  operation,  and  reduces  the  result  to  the  lowest 
terms. 

5.  Eeduce  £  of  J  of  %  to  a  simple  fraction.    Ans.  -fa. 

Reduce  the  following  to  simple  fractions : 

6.  I  of  }  of  f.       11.  J  off  off        16.  I  off  of  11. 

7.  f  of  i  of  |.       12.  i  of  J  of  4|.       17.  i  of  i  of  6J. 

8.  J  off  of  i.       13.  f  of  I  of  A.       18.  I  off  of  9. 

9.  f  of  i  off.        14.  i  of  *  off         19.  J  off  of  2*. 
10.  I  of  i  off.       15.  fofifofi.      20.  J  off  of  10*. 


CASE    VII. 
To  reduce  a  Fraction  to  any  acquired  Denominator. 

1.  Reduce  %  to  twelfths. 

Analysis. — The  given  denominator  4  is  Operation. 

contained  in  12,  the  required  denominator,  3  12-^4  =  3. 

times  ;  therefore,  multiplying  both  terms  of  3  x  3 

5  by  3,  it  becomes  rif  and  is  the  fraction  re-  7  x  ~  =  TJ>  -^W* 
quired. 


FRACTIONS.  •  IGkl 

27.  How  reduce  a  fraction  to  any  required  denominator  ? 
Multiply  both  terms  of  the  fraction  by  such  a  number 

as  will  make  the  given  denominator  equal  to  the  required 
denominator.     (P.  95,  Q.  20,  n.) 

Beduce  the  following  fractions  to  the  denominators 
indicated : 

2.  f  to  3oths.      5.  -fa  to  72ds.        8.  ff  to  i7ists. 

3.  I  to  4oths.       6.  H  to  99ths.        9.  f§  to  2  76ths. 

4.  f  to  35ths.       7.  H  to  i44ths.    10.  -Jgg  to  ioooths. 

CASE    VIII. 
To  reduce  Fractions  to  a  Common  Denominator. 

28.  What  is  a  Common  Denominator  ? 

A  Common  Denominator  is  one  that  belongs 
equally  to  two  or  more  fractions  ;  as,  f ,  f,  f . 

1.  It  is  required  to  reduce  \,  \,  and  },  to  equivalent 
fractions,  having  a  common  denominator. 

Analysts. — If  we  multiply  each  denom-  Operation. 

inator  by  all  the  other  denominators,  the  1x3x4 

several  products  will  be  the  same  ;  for,     1     2x3x4 —  ^ 4' 
each  is  composed  of  the  same  factors,  2,  3, 

and  4,  the  product  of  which  is  24.    Again,      i  — 7— _^.. 

if  we  multiply  each  numerator  by  all  the  3  x  2  x  4 

denominators  except   its  own,  it  follows      x 1x2x3 6 

that  the  terms  of  each  fraction  will  be  mul-     *     4x2x3     ~^*' 

tiplied   by  the   same  number ;   therefore, 

the  value  of  the  fractions  is  not  altered.    (P.  93,  Prin.  V.) 

20.  How  reduce  fractions  to  a  common  denominator  ? 

Multiply  the  terms  of  each  fraction  by  all  the  denomi- 
nators except  its  oivn. 

Notes. — 1.  Mixed  numbers  must  be  reduced  to  improper  frac- 
tions, and  compound  fractions  to  simple  ones,  before  applying  the 
rule.    (Ex.  12.) 


10£  ItE'DUCTION     OP 

2.  It  should  be  observed  that  the  value  of  the  given  fractions 
is  not  altered  by  reducing  them  to  a  common  denominator.  The 
reason  is,  that  the  terms  of  each  fraction  are  multiplied  by  the 
eame  numbers.     (P.  93,  Prin.  V.) 

2.  Eeduce  -f,  J,  ^,  to  a  common  denominator. 
Reduce  the  following  to  a  common  denominator : 


3.  f  and  £. 

6.  1,  i  i 

9-   TT>  &  f 

4.  f  and  \. 

7.  f,  i  f 

10.  H,  }*, ** 

5.  |  and  f . 

Q      2     2     7 

°*    "9"'  ~3~>  ~5' 

TT       19       SO         7 
XI'    TT?  T(JOJ  TH 

12.  Find  a  common  denominator  of  |  of  f,  3f  and  5. 

Analysis. — \  of.  f=f;  3%—Lt,  and  5=4.    Now,  f,  V",  and  \ 
by  the  rule,  become  -2a4-,  f  |, J^1. 

13.  Reduce  *j,  J  of  f ,  and  3  to  a  common  denomi- 
nator. 

14.  Reduce  -f  of  f,  7,  and  51  to  a  common  denomi- 
nator. 

CASE    IX. 

To  reduce  Fractions  to  the  Least  Common  Denominator. 

SO.  What  is  the  Least  Common  Denominator  of  two  or  more 
fractions. 

The  Least  Common  Denominator  of  two  or 

more  fractions  is  the  least  common  multiple  of  their  de- 
nominators.    (P.  86,  Q.  17.) 

1.  What  is  the  least  common  denominator  of  -f,  fa 
and  T\  ? 

Analysis. — The  solution  of   this  ,  \,    Tn    TC 

and  similar  Examples  requires  two 

steps :  1st.  To  find  the  least  com.  mul-  5  )  J>   IO>     5 

tiplc  of  the  denominators  for  the  re-  "^       ~2       Y. 

quired  denominator.  2d.To  reduce  the         »/-•*>«      r    n  is 

given  fractions  to  this  denominator.  u     J 


FRACTIONS.  103 

The  least  com.  multiple  of  3, 10,  and  2X10 

15  is  30.     (P.  87.)    To  reduce  f,  A,  3  x  io^T°' 

and  -,A5    to    thirtieths,   we    multiply 

both  terms  of  the  given  fractions  by  —        =^.  f-.^?w. 

such  a  number  as  will  reduce  them  ■•' 

to  thirtieths.     Now  multiplying  both  _4_  x  2__ .  8 

terms  of  f  by  10,  the  fraction  becomes  15x2 

IS  ;  multiplying  both  terms  of  *fo  by 

3,  the  fraction  becomes  &  ;  and  multiplying  both  terms  of  W  b, 
2,  we  have  3a0-.  (P.  101,  Q.  27.)  Therefore,  the  required  fractions 
are  §#,  -,90-,  and  -380-. 

31.  How  reduce  fractions  to  the  least  common  denominator  ? 

Find  the  least  common  multiple  of  all  the  denomina- 
tors; then  multiply  both  terms  of  each  fraction  by  such 
a  number  as  will  reduce  it  to  this  denominator.  (P.  10 1, 
Q.27.) 

Note. — Mixed  numbers  must  be  reduced  to  improper  fractions, 
compound  fractions  to  simple  ones,  and  all  fractions  to  their  lowest 
terms,  before  applying  the  rule.    (Ex.  12.) 

2.  Eeduce  -J,  -J,  and  J  to  the  least  com.  denominator. 
Solution. — The  least  com.  multiple  of  2,  3,  and  4  is  12.    Now 
i=fV ;  *=-& ;  and  |_=-&.    Ans.  -&,  -fr,  -,%-. 

Eeduce  the  following  to  the  least  com.  denominator: 
2    2    3    I  6    3    2.     7  o     -4     A    ?3 

4.  h  h  tV  7-  A>  i  A-  i°-  T&  i  y>  £• 

r-       7       5       I  Q      8      6      2  tt       A      JL     .5       ?- 

1 2.  Find  the  least  com.  denominator  of  2J,  £  of  ^ 
J  and  5. 

Analysis.— 2*=V- ;  £  of  ft-ft^rVj  f=i  *-*  5=t-  The 
least  com.  multiple  of  5,  10,  4,  and  1,  is  20.    Now  L6L—tt'>  iV=Ai 

2  — 15..    on(l  3_ljQ.Il         JOTo   4.1    JL     15.    Ilia 

13.  What  is  the  least  com.  denominator  of  §,  5 J, 
and  I  ? 

14.  What  is  the  least  com.  denominator  of  J  of  f,  4}, 
I,  and  4  ? 


i.04  ADDITION      OF     FRACTION'S. 

ADDITION     OF     FRACTIONS. 

To  add  Fractions  which  have  a  Common  Denominator. 

Remark. — When  two  or  more  fractions  express  parts  of  the 
same  kind  of  unit,  and  have  a  common  denominator,  their  nu- 
merators are  like  numbers  ;  hence,  they  may  be  added,  subtracted, 
and  divided  as  whole  numbers. 

MENTAL    EXERCISES. 
i.  "What  is  the  sum  of  f-  dollar,  f-  dollar,  and  -f  dollar  ? 

Analysis.— 3  fifths  dollar  and  2  fifths  are  5  fifths,  and  4  are 
9  fifths,  which  are  equal  to  i£  dollar. 

2.  What  is  the  sum  of  -J,  f,  and  \  ? 

3.  What  is  the  sum  of  f ,  J,  f,  and  £  ? 

4.  What  is  the  sum  of  |,  f ,  f ,  and  -f  ? 

5.  What  is  the  sum  of  f,  -J,  |,  and  |  ? 

6.  What  is  the  sum  of  T33-,  -f^,  yV,  and  T8T  ? 

.      SLATE     EXERCISES. 

32.  When  fractions  have  a  common  denominator,  what  is  true 
of  their  numerators  ? 

Their  numerators  express  like  parts  of  a  unit,  and 
therefore  are  like  numbers. 

1.  What  is  the  sum  of  J$,  -££,  and  \  %  ? 

Analysis. — As    these  fractions  Operation. 

have  a  common  denominator,  their  ~zis  +  ^u  +  To"  —  "2To~>  or  2"2o~* 
numerators     are    like    numbers. 

Hence,  they  may  be  added  as  whole  numbers.  (P.  25.  Q.  9.) 
Thus,  the  sum  of  13  twentieths +  16  twentieths  + 14  twentieths= 
$1,  or  2fij,  Ans. 

33.  How  add  fractions  which  have  a  common  denominator? 
Add  the  numerators,  and  place  the  sum  over  the  common 

denominator. 

Note. — The  answers  shculd  be  reduced  to  the  l-zoest  terms, 
and  improper  fractions  to  whole  or  mixed  number*. 


ADDITION      OF      FRACTIONS.  105 

2.  A  man  sold  f  of  an  acre  of  land  to  one  customer, 
J  to  another,  f  to  another,  and  £  to  another :  how  much 
did  he  sell  to  all  ? 

3.  What  is  the  sum  of  ^  pound,  -^  pound,  jj  pound, 
and  ^  pound  ? 

4.  What  is  the  sum  of  -^  -JJ,  if,  £J  ? 

5.  What  is  the  sum  of  Jf,  A>  tt  ft  ? 

6.  What  is  the  sum  of  ■&,  f  J,  j|,  jj  ? 

To  add  Fractions  which  have  Different  Denominators. 

1.  What  is  the  sum  of  J  dollar,  £  dollar,  and  £  dollar  ? 

Analysis. — Since   these    frac-  Operation. 

tions  have  different  denominators,  2  X  4  X  6  =  48    Cow.  Z). 

tlieir  numerators  denote  unlike         ,  v,  .  v^  <  _  „  .    T  „/   at 

1x4x0  =  24,  1  Sf  Jy. 
parts  of   a  unit;    consequently,  a—    a       7  aj 

they  cannot  be  added,  any  more         3X2x6-3°>  2«  ^ 
than  units  of  different  orders.  5  x  2  X  4=40,  3  J  JV. 

We  therefore  reduce  them  to  a     24 -_|_  3_6  _|_4o_ J£-0-f  or  2-^. 
common  denominator,  which  is 
48,  and  add  the  numerators,  as  above. 

Or,  we  may  reduce  the  fractions  to  the  least  common  denomi- 
nator, which  is  12,  and  then  add  the  numerators.  Thus,  i=  £>  ;  $ 
=  ft  ;  and .§=!§• :  now  ft  +  &  +if =li,  or  2-^,  the  same  as  before. 

84.  What  then  is  the  general  rule  for  adding  fractions  ? 

Reduce  tlmn  to  a  common  denominator,  and  place  the 
sum  of  the  numerators  over  it. 

Or,  reduce  them  to  the  least  common  denominator,  and 
over  this,  place  the  sum  of  the  numerators. 

Note. — The  integral  and  fractional  parts  of  mixed  numbers 
should  be  added  separately,  and  the  results  be  united.    (Ex.  12.) 

Or,  mixed  numbers  may  be  reduced  to  improper  fractions,  and 
compound  fractions  to  simple  ones,  and  then  be  added.    (Ex.  18.) 

2.  Henry  paid  $  dollar  for  an  arithmetic,  }  dollar  for 
a  slate,  and  £  dollar  for  a  geography :  what  did  he  pay 
for  all  ?  Ans.  1  ^  dol. 


106  ADDITION      OF      FRACTIONS. 

3.  What  is  the  sum  of  |  pound,  £  pound,  and  ^ 
pound  ? 

4.  Add  f,  J,  and  |.  8.  Add  TVf,  and  f. 

5.  Add  f,  f,  and  i  9.  Add  ^,  Jj,  and  J-J. 

6.  Add  ^  -J,  and  }f  10.  Add  Jfc  ^  and  f  J. 

7.  Add  A,  ^,  and  ft  1 1.  Add  Jf,  JJ,  and  f  f . 

12.  What  is  the  sum  of  10J  pounds,  17  J,  and  2oS 
pounds  ? 

Analysis. — Reducing  the  fractional  parts  to  ioi^io-1-0-. 

the  least  common  denominator,  which  is  40,  we  I73_I724 

add  the  fractions  and  integers  separately.  5 JY" 

The  sum  of  the  fractions  is  $f=lif.    Adding  23?— 23?tr- 

the  1  to  the  whole  number,  the  sum  is  51^0,  Ans.  Ans.  5 1 3(7. 

13.  How  many  pounds  of  tea  are  there  in  2  chests, 
containing  45-J  and  56-^  pounds  respectively  ? 

14.  How  much  cloth  in  3  pieces,  containing  12  J,  17}, 
and  2 if  yards? 

15.  If  a  man  walks  2of  miles  in  one  day,  25  J  the  next, 
and  31!  the  next,  how  far  will  he  travel  in  all  ? 

16.  If  a  housekeeper  buys  3  J  dollars  worth  of  sugar, 
5!  dollars  worth  of  coffee,  and  15I  dollars  worth  of  flour, 
what  is  the  amount  of  her  bill  ? 

17.  Three  men  buying  a  sail-boat,  put  in  2 7  J,  23I,  and 
2of  dollars  respectively:  what  was  the  cost  of  the  boat? 

18.  What  is  the  sum  of  £  of  f,  \  of  ■&,  and  J  of  2  J  ? 

Solution.— Reducing  the  compound  fractions  to  simple  ones, 
we  have  \  of  £=i,  \  of  /o=ViT,  and  £  of  2£=£.  Reducing  these 
fractions  to  the  least  common  denominator,  20,  they  become  A,, 
2Lif,  and  fft;  and  the  sum  tfr+^+M=Hi  or  i\,  Ans. 

19.  What  is  the  sum  of  f  of  J,  J  of  ft,  and  5  J  ? 

20.  Add  f  of  45,  tt  of  A>  and  ^  of  3}. 

21.  Add  2  2|,  J  off,  and  £  of  ~fa. 

22.  What  is  the  sum  of  -f  of  4.  28  J,  and  15}  ? 


SUBTRACTION      OF      FRACTIONS.  107 

SUBTRACTION     OF     FRACTIONS. 
To  Subtract  Fractions  which  have  a  Common  Denominator. 

i.  If  Frank  has  f  of  a  pound  of  maple  sugar,  and 
gives  away  f  of  it,  how  much  will  he  have  left  ? 

Analysis.— 3  fifths  from  4  fifths  leaves  1  fifth.    Therefore,  he 
will  have  1  fifth  of  a  pound  left. 

2.  From  i  yard,  take  .$■  yard  ? 

3.  What  is  the  difference  between  ^  of  a  dime  and 
•^  of  a  dime  ? 

4.  What  is  the  difference  between  ■&  and  ^  ? 

5.  What  is  the  difference  between  ^  of  a  week  and  f 
of  a  week  ? 

6.  What  is  the  difference  between  T5^-  and  -fj  ? 

SLATE     EXERCISES. 

1.  What  is  the  difference  between  ~fe  of  a  foot  and 
-f-J  of  a  foot  ? 

Analysis. — Since  these  fractions  have  a         Operation. 
common  denominator,  their  numerators  are      TI     TZ— TZ  It. 
like  numbers,  and  may  be  subtracted  as  whole 
numbers.    (P.  104,  Rem.)    ||  minus  -fe  equal  -fe  foot,  Ans. 

35.  How  subtract  fractions  which  have  a  common  denominator? 
Take  the  less  numerator  from  the  greater,  and  place  the 
difference  over  the  common  denominator. 

2.  From  H  of  a  day,  subtract  \\  of  a  day.    Ans.  -}  d. 

3.  From  !  J  °f  a  ton,  subtract  Jf  of  a  ton. 

4.  From  |-|  of  a  bushel,  subtract  f-|  of  a  bushel. 

5.  From  iJf,  subtract  ^. 

6.  From  AtV?  subtract  T3^rV 

7.  What  is  the  difference  between  ^-  and  ^/o"  -? 

8.  What  is  the  difference  between  |ffi  and  fff-f  ? 


108  SUBTRACTION      OF      FRACTIONS. 

To  Subtract  Fractions  which  have  Different  Denominators. 

i.  It  is  required  to  find  the  difference  between  | 
and  j$. 

Analysis. — Since  these  fractions  have  Operation. 

not  a  common  denominator,  their  nu-       8x12  =  96,  CD. 
n levators  are  unlike  numbers;    conse-     3_ 36.  an(j  10  — 8o# 
quently   one  cannot  be  taken  directly      8o_3  6  — 44       .  iV 
from  the  other.     Hence,  we  reduce  them     ^     ^     ^^,        *** 
to  a  common  denominator,  and  subtract  as  above. 

36.  What  then  is  the  general  Rule  for  subtracting  fractions  ? 
Reduce  them  to  a  common  denominator,  and  over  it 
place  the  difference  of  the  numerators. 

Notes. — 1.  WJwle  and  mixed  numbers  should  be  reduced  to 
improper  fractions,  and  compound  fractions  to  simple  ones ;  then 
proceed  as  above.    (Ex.  15,  23.) 

2.  If  both  are  mixed  numbers,  it  is  sometimes  more  expeditious 
to  reduce  the  fractions  to  a  common  denominator ;  then  subtract 
the  fractional  and  integral  parts  separately.     (Ex.  17.) 

3.  The  operation  may  often  be  shortened  by  reducing  the  frac- 
tions to  the  least  common  denominator. 

2.  Bought  a  cargo  of  corn,  at  |  of  a  dollar  a  busheb 
and  sold  it  at  f  of  a  dollar:  what  was  the  gain  \ei' 
bushel  ? 

'     3.  A  man  owning  -J-f  of  a  ship,  sold  J  of  her :  wl  .at 
part  had  he  left  ? 

4.  From  -J,  take  f  8.  From  f  J,  take  f . 

5.  From  \\,  take  f.  9.  From  fj,  take  §. 

6.  From  Jf,  take  £J.  10.  From  ££,  take  £{, 

7.  From  3f,  take  ■&.  1 1.  From  ^,  take  $} 

1 2.  What  is  the  difference  between  ^ff  and  -f£T  ? 

13.  What  is  the  difference  between  %%%  and  %£6  ? 

14.  What  is  the  difference  between  £ Jg  and  f-J2  ? 


SUBTKACTIOtf      OF  FEACTIOKS.  109 

15.  Subtract  14-f  hogsheads  from  36  hogsheads  ? 

Analysis. — Reducing  36  and  14^  1st  Operation. 

to  improper  fractions,  we  have  Ha  36=— f2-. 

and x^a.    Now 252  minus  103 equals  j^s—LQ^ 

x?- ;  and  -4a  equals  aif«    -4.7W.  2if  2 - 2  _  f  Q   _  1 49.  or  2  j  a 
hogsheads. 


2d  Operation. 

36 


Or,  borrowing  1,  which  equals 
\9  we  have  \  minus  f,  equal  to  f . 
Then  1  to  carry  to  14  makes  15  ;  *4y 

and  36  minus  15=21.    Ans.  2if.  Ans.  21^  h. 


16.  A  farmer  having  85  bushels  of  wheat,  sold  63I 
bushels :  how  much  had  he  left  ? 

17.  What  is  the  difference  between  21  j  tons  and  i6f 
tons  ? 

Analysis. — Reducing  the  fractions  to  the  Operation. 

common  denominator  15,  the  minuend  2i|     2lf=2i^f. 
=2i}£;  and  the  subtrahend  16?;= i6}f.  Now     .gi_.  ,51a 

j§  is  larger  than  j§,  the  fraction  above  it ;  5 _5 

hence  we  borrow  1,  or  \§,  and  add  it  to  |f,        Ans.  4^-f  tons, 
making  fjf;  and  ff— ||=|f ;  carrying  1  to 
16  makes  17,  and  21  —  17=4.    Ans.  4}!  tons. 

18.  From  a  cask  of  molasses,  containing  56!  gallons, 
20  }  gallons  were  drawn :  how  many  remained  ? 

19.  Take  18  j  from  37 \.        21.  Take  62  }  from  83-J-. 

20.  Take  31J  from  66%.        22.  Take  106J  from  135^. 

23.  Required  the  difference  between  J  of  f  of  5,  and 

i  off  of  3|. 

Solution. — Reducing  the  compound  fractions  to  simple  ones, 
cancelling,  etc.,  the  minuend  becomes  §,  and  the  subtrahend  f. 
Again,  reducing  f  and  f  to  the  common  denominator  15,  we  ob- 
tain if,  and  fr;  and  fg- -&=-&,  4««. 

24.  From  f  of  J,  take  -J  of  f . 

25.  From  |-  of  T\,  take  -^  of  T3T. 


110      MULTIPLICATION      OF      FBACTIONS. 

MULTIPLICATION    OF    FRACTIONS. 

CASE    I. 

To  Multiply  a  Fraction  by  a  Whole  Number. 

i.  At  \  cent  apiece,  what  will  3  plums  cost  ? 

Analysis. — Since  1  plum  costs  \  cent,  3  plums  will  cost  3 
times  as  much  ;  and  3  times  1  half  are  3  halves,  equal  to  i^  cent. 
Therefore,  3  plums  will  cost  1  \  cent. 

2.  At  J  of  a  dollar  a  pound,  what  will  3  pounds  of 
tea  come  to  ? 

3.  If  a  lad  earn  j  of  a  dollar  per  day,  how  much  will 
he  earn  in  5  days  ? 

4.  What  cost  6  bushels  of  apples,  at  |  of  a  dollar  a 
bushel  ? 

5.  What  is  the  product  of  5  times  T6T  ?    Of  7  times  J  ? 

6.  What  cost  12  photographs,  at  §  dollar  apiece  ? 

7.  What  cost  1 1  rabbits,  at  f  dollar  apiece  ? 

8.  What  cost  5  oranges,  at  d\  cents  each  ? 
Analysis. — If  1  orange  costs  6^  cents,  5  will  cost  5  times  6^ 

cents.    Now,  5  times  6  cents  are  30  cents,  and  5  times  \  are  $ , 
equal  to  i^,  which  added  to  30  make  31^  cents.    Therefore,  etc. 

9.  At  2>\  dimes  each,  what  will  9  melons  cost  ? 
10.  What  cost  12  gold  pens,  at  4!  dollars  apiece  ? 

SLATE      EXERCISES. 
1.  At  I  dollar  a  box,  what  will  3  boxes  of  starch  cost  ? 

Analysis. — Since  1  box  costs  {j-  dollar,  1st  Operation 

3  boxes  will  cost  3  times  {j-  dol. ;  and  3     |.  x  3^=-^,  or  2 \  d 
times  |=  Lb£,  or  2$  dollars,  the  cost  required. 

Or,  if  we  divide  the  denominator  by  3,  2d  Operation. 

the  result  will  be  the  same;  for,  diviling     J— ^-3  =  ^  or  2\  d. 
the  denominator  multiplies  the  fraction. 

37.  How  multiply  a  fraction  by  a  whole  number? 

Multiply  the  numerator  by  the  whole  number. 

Or,  divide  the  denominator  by  it.    (P.  93,  Prin.  IV.) 


MULTIPLICATION      OF      FRACTIONS.       Ill 

Notes. — i.  The  second  method  is  preferable,  when  the  denom- 
inator can  be  divided  by  the  whole  number  without  a  remainder. 

2.  If  the  multiplicand  is  a  mixed  number,  multiply  the  frac- 
tional and  integral  parts  separately,  and  unite  the  products. 

Or,  reduce  the  mixed  number  to  an  improper  fraction ;  then 
apply  the  rule.    (Ex.  15.) 

2.  What  will  1 7  pounds  of  honey  cost,  at  £  of  a  dollar 
a  pound  ? 

3.  What  cost  25  bushels  of  potatoes,  at  T5^  of  a  dollar 
a  bushel  ? 

4.  At  4^  dollars  apiece,  what  will  3$  straw  hats 
come  to  ? 

5.  Multiply  jj  by  15.  10.  Multiply  \\  by  26. 

6.  Multiply  \%  by  17.  11.  Multiply  ||  by  42. 

7.  Multiply  |J  by  9.  12.  Multiply  fj  by  50. 

8.  Multiply  ff  by  11.  13.  Multiply  ffo  by  83. 

9.  Multiply  If  by  18.  14.  Multiply  ^  by  no. 

15.  What  will  5  hundred  weight  of  sugar  cost,  at  6  J 
dollars  per  hundred  ? 

Analysis. — Multiplying  the  fraction  and  inte-  Operation. 
ger  separately  by  the  whole  number,  we  have  J  x  6|.  dols. 

5=-^,  or  Jf  j    and  6x5=30.     Now  30+3l=33i         5 
dols.    Therefore,  etc.  ~y  ,  , 

Or,  the  mixed  number  6}=*£,  and  ¥x$=*f*  33* 
=334  dollars,  the  cost  required. 

16.  What  cost  23  yards  of  muslin,  at  12}  cents  a  yard  ? 

17.  What  cost  45  yearlings,  at  18 J  dollars  apiece  ? 

18.  Multiply  31 J  by  25.        21.  Multiply  62^-  by  57. 

19.  Multiply  37^  by  42.        22.  Multiply  66f  by  75. 

20.  Multiply  40!  by  61.       23.  Multiply  87^  by  100. 

24.  What  cost  24  bureaus,  at  27 J  dollars  apiece  ? 

25.  What  cost  12  melodeons,  at  62-J  dollars  apiece  ? 

26.  What  cost  31  sofas,  at  71 J  dollars  apiece  ? 


112       MULTIPLICATION      OB      Fit  ACTIONS. 

CASE     II. 
To  Multiply  a  Whole  Number  by  a  Fraction. 

i.  What  will  J  yard  of  edging  cost,  at  10  cents  a  yard  ? 
Analysis. — If  2  halves,  or  a  whole  yard,  cost  10  cents,  1  half 
yard  will  cost  1  half  of  10  cts. ;  which  is  5  cts.     Therefore,  etc. 

.2.  If  a  dozen  eggs  cost  16  cts.,  what  will  \  dozen  cost? 

3.  If  a  melon  is  worth  12  cents,  what  is  \  of  it  worth  ? 

4.  If  a  pie  is  worth  20  cents,  what  is  J  of  it  worth? 

5.  What  cost  f  pound  of  grapes,  at  14  cents  a  pound  ? 
Analysis. — Since  1  pound  is  worth  14  cents,  §  of  a  pound  are 

worth  %  of  14  cents.    But  1  third  of  14  cents  is  45-  cents,  and  2 
thirds  are  2  times  4I,  or  9^  cents.     Therefore,  etc. 

6.  If  a  cake  costs  80  cents,  what  will  J  of  it  cost  ? 

SLATE     EXERCISES. 
3§.  What  is  meant  by  multiplying  by  a  fraction? 

Multiplying  by  a  Fraction  is  taking  a  certain 
part  of  the  multiplicand  as  many  times  as  there  are  Ufa 
parts  of  a  unit  in  the  multiplier. 

39.  How  find  a  fractional  part  of  a  number  ? 

Divide  the  number  into  as  many  equal  parts  as  there 
are  units  in  the  denominator,  and  then  take  as  many  of 
these  parts  as  there  are  units  in  the  numerator.    That  is, 

To  multiply  a  number  by  J,  divide  it  by  2. 

To  multiply  a  number  by  J,  divide  it  by  3. 

To  multiply  a  number  by  J,  divide  it  by  4  for  J,  and 
multiply  this  quotient  by  3  for  J,  etc. 

Remarks. — 1.  Multiplying  a  whole  number  by  a  fraction  is  the 
same  as  taking  a  corresponding;  fractional  part  of  the  number. 

2.  When  the  multiplier  is  1,  the  product  is  equal  to  the  mulfi- 
plicand;  when  the  multiplier  is  greater  than  1,  the  product  is 
greater  than  the  multiplicand  ;  when  the  multiplier  is  less  than 
1,  the  product  is  less  than  the  multiplicand. 


MULTIPLICATION      OF      FRACTIONS.       113 


:.  What  will  f  of  a  gallon  of  cider  cost,  at  38  cents  a 
gallon  ? 

Analysis. — Since  1  gallon  costs  38  cents, 
§  of  a  gallon  must  cost  §  times  38.  or  f  of  38 
cts.  Now  ^  of  38  cts.  is  12  J  cts.,  and  2  thirds 
are  2  times  I2f  cts.  Multiplying  I2|  by  2, 
we  have  2  times  f=4,  or  1^.  2  times  12  are 
24,  and  1^  make  25^  cts.,  the  cost  required. 

Or,  thus:  §  of  a  gallon  will  cost  i  of  2 
times  the  cost  of  1  gallon.  Now  2  times  38 
cts.  are  76  cts.,  and  ^  of  76  cts.  equals  76-7-3, 
or  25^  cts.,  the  same  as  before. 


1st  Operation. 

3)38 
"I 

2 

Ans.  25J  cts. 

2d  Operation. 
38  X  2  =  76. 

76-^3=25^  cts. 


40.  How  multiply  a  wlwle  number  by  a  fraction? 

Divide  tlie  wlwle  number  by  the  denominator  of  tlie 
fraction,  and  multiply  by  the  numerator. 

Or,  multijrty  the  whole  number  by  the  numerator  of 
the  fraction,  and  divide  by  the  denominator. 

Notes. — 1.  The  fraction  may  be  taken  for  the  multiplicand, 
and  the  whole  number  for  the  multiplier,  at  pleasure,  without 
affecting  the  result.    (P.  47,  Rem.) 

2.  Mixed  numbers,  when  multipliers,  may  be  reduced  to  im- 
proper fractions ;  then  proceed  according  to  the  rule. 

Or,  multiply  by  the  fractional  and  integral  parts  separately, 
and  unite  the  results. 

2.  If  a  bushel  of  barley  is  worth  75  cts.,  what  is  f  of  a 
bushel  worth  ? 

3.  If  an  acre  of  land  is  worth  $100,  what  is  f  of  aq 
acre  worth  ? 


4.  Multiply  45  by  %. 

5.  Multiply  61  by  f. 

6.  Multiply  78  by  f 

7.  Multiply  87  by  f. 


8.  Multiply  no  by  T3T. 

9.  Multiply  238  by  ^. 
10.  Multiply  378  by  {f 
n.  Multiply  500  by  -fi^. 


12.  In  1  year  there  are  365  days:  how  many  days  are 
there  iu  f  of  a  year  ? 


Operation. 

)42      • 

dols. 

I 

ton, 

H 

252 

M 

6 

a 

8f 

« 

t 

u 

25i 

U 

J 

a 

114       MULTIPLICATION      OF      FRACTIONS. 

13.  What  cost  6f  tons  of  iron,  at  42  dollars  a  ton? 

Analysis. — If  1  ton  costs  42  dol- 
lars, 6}  tons  will  cost  6i  times  42  dols. 
We  first  multiply  by  the  whole  num- 
ber 6,  and  the  product  is  252.  In  mul- 
tiplying by  the  fraction  £,  we  take  £  of 
the  multiplicand,  and  setting  it  under 
the  product  of  the  integral  part,  mul- 
tiply it  by  3;  for,  fc=i+f.  We  now 
have  the  partial  products  of  6,  of  £,  and  285!  dols.  6f  tons, 

f,  and  their  sum,  285!  dols.,  is  the 

answer  required. 

14.  What  cost  8 J  yards  of  alpaca,  at  80  cts.  a  yard  ? 

15.  If  a  man  can  walk  45  miles  a  day,  how  far  can  he 
walk  in  iof  days? 

16.  Multiply  52  by  6J.  19.  Multiply  101  by  10-i. 

17.  Multiply  57  by  7  J.  20.  Multiply  365  by  n|. 

18.  Multiply  78  by  8 J.  at.  Multiply  500  by  12 J. 


CASE    III. 
To  Multiply  a  Fraction  by  a  Fraction. 

1.  What  will  A  of  a  gallon  of  syrup  cost,  at  £  of  a 
dollar  a  gallon  ? 

Analysis. — 1  tenth  of  a  gallon  will  Operation. 

cost  fa  the  price  of  1  gal. ;  and  -fa  of  £         5      4*  2  _ 
dol.  is  ft  dol.     (P.  93,  Prin.  III.)  3,&Xis>~~T 

Again,  -fa  gal.  will  cost  4 times  as  much 
as  -,1,,  ;  and  4  times  fa  are  gft,  or  $  dol.    In  the  operation  we  cancel 
the  common  factors,  and  multiply  the  numerators  together,  and 
then  the  denominators. 

41.  How  multiply  a  fraction  by  a  fraction? 
Cancel  the  common  factors  ;  then  multiply  the  numera- 
tors together  for  the  new  numerator,  and  the  denominators 

for  the  new  denominator. 


MULTIPLICATION      OF      FEACTIOKS.       115 

Notes. — i.  Compound  fractions  are  multiplied  like  simple  ones ; 
the  word  of  being  equivalent  to  the  sign  x  . 

2.  Reduce  mixed  numbers   to  improper  fractions,  and  then 
multiply  them  according  to  the  rule.     (Ex.  16.) 

3.  The  object  in  cancelling  the  common  factors  is  twofold :  it 
x?iortens  the  operation,  and  gives  the  answer  in  the  lowest  terms. 

2.  What  will  f  of  a  pound  of  sugar  cost,  at  -J  of  a 
dollar  a  pound  ? 

3.  What  cost  J  of  a  yard  of  muslin,  at  t2q  of  a  dollar 
a  yard  ? 


4.  Multiply  f  by  J. 

10.  Multiply  f  by  J  x  f. 

5.  Multiply  f  by  TV 

11.  Multiply  f  by  £x^. 

6.  Multiply  T\  by  J. 

12.  Multiply  f  by  J  x -ft. 

7.  Multiply  tV  by  f 

13.  Multiply  TV  x  A  x  -2T- 

8.  Multiply  }f  by  fj. 

14.  Multiply  Jxfx f  Xf. 

9.  Multiply  If  by  f|.  15.  Multiply  J  x  J  x  ^  x  }f 
16.  What  cost  8£  yards  of  calico,  at  1 2\  cents  a  yard  ? 
Solution— 81=^ ;  i*i=¥.    Now>  ¥  x  ¥=&iA,  or  109J  cts. 

4.£.  The  preceding  principles  may  be  summed  up  in 
the  following 

GENERAL  RULE. 

Reduce  wlwle  arid  mixed  numbers  to  improper  frac- 
tions; tlien  cancel  the  common  factors,  and  place  the  pro- 
duct of  the  numerators  over  the  product  of  the  denom- 
inators. 

EXAMPLES    FOR    PRACTICE. 

1.  What  is  the  product  of  fxfxfxlf? 

2.  Multiply  j  of  i  of  1 J  by  f  of  T%. 

3.  Multiply  f  of  f  of  ^  by  §  of  4J. 

4.  Multiply  f  of  6\  by  |  of  ||  of  8. 

5.  Multiply  f  of  J  of  18  by  f  of  25. 

6.  What  is  the  product  of  6£  multiplied  by  2f  ? 


116  DIVISION      OF      FRACTIONS. 

7.  What  cost  10 }  pounds  of  beef,  at  15^  cents  a  pound  ? 

8.  At  1  if  dollars  a  barrel,  what  will  20 \  barrels  of  vin- 
egar come  to  ? 

9.  Multiply  i6|  by  g£.      12.  Multiply  45!  by  31  J. 

10.  Multiply  31 J  by  1 8 J.    13.  Multiply  66|  by  3 7  J. 

11.  Multiply  37 J  by  i6|.    14.  Multiply  110J  by  60^. 


DIVISION    OF    FRACTIONS. 

CASE    I. 

To  Divide  a  Fraction  by  a  Whole  Number. 

1.  If  2  citrons  cost  T\  of  a  dollar,  what  will  1  citron 
cost? 

Analysis. — If  2  citrons  cost  -fo  of  a  dollar,  1  will  cost  1  half  of 
/tf  of  a  dollar ;  and  1  half  of  4  tenths  is  -fc,  or  £  of  a  dollar. 

2.  If  3  apples  cost  -^  of  a  dime,  what  will  1  apple 
cost? 

3.  If  4  peaches  cost  T8^  of  a  shilling,  what  will  1  cost  ? 

4.  If  5  yards  of  calico  cost  -f  £  of  a  dollar,  what  will  1 
yard  cost  ? 

5.  If  3  doves  cost  |  dollar,  what  will  1  dove  cost  ? 

Analysis. — 1  dove  is  ^  of  3  doves ;  therefore,  1  dove  will  cost 
£  of  i  dollar,  and  £  of  £  dollar  equals  £  dollar.     (P.  63,  Q.  12.) 

6.  If  4  balls  cost  -J  dollar,  what  will  1  ball  cost  ? 

7.  If  J  bushel  of  oats  are  equally  divided  among  5 
horses,  how  many  will  each  horse  receive  ? 

8.  If  I  of  a  barrel  of  apples  are  divided  equally  among 
7  persons,  what  part  of  a  barrel  will  each  receive  ? 

9.  If  f-$  of  an  acre  of  land  are  divided  into  4  equal 
lots,  how  much  will  there  be  in  each  lot  ? 

10.  If  \%  of  a  ton  of  hay  are  divided  into  3  equal 
loads,  how  much  will  there  be  in  each  load  ? 


DIVISION"     OF     FRACTIONS 


117 


1st  Operation. 
2d  Operation. 


3=A- 


Ans. 


A= 


|dol. 


SLATE    EXERCISES. 

i.  If  3  pounds  of  raisins  cost  f  dollar,  what  will 
pound  cost  ? 

1st  Method.— i  pound  is  £  of  3  pounds, 
therefore  1  pound  will  cost  £  of  f  dol.  Di- 
viding the  numerator  into  3  equal  parts,  we 
have  f  dol.-f-3  =  f  dollar.     (P.  93,  Prin.  II.) 

2d  METnOD. — Since  multiplying  the  de- 
nominator divides  a  fraction,  it  follows  that 
$  dol.-^-3=/r,  or  f  dol.,  the  same  as  before. 
(P.  93,  Prin.  III.) 

Remahk. — The  solution  of  this  and  similar  examples  is  an 
application  of  the  second  office  of  Division.     (P.  63,  Q.  10.) 

43.  How  divide  &  fraction  by  a  whole  number? 
Divide  the  numerator  by  the  whole  number. 
Or*  multiply  the  denominator  by  it. 

Notes. — 1.  When  the  dividend  is  a  mixed  number,  reduce  it 
to  an  improper  fraction ;  then  apply  the  rule.    (Ex.  13.) 

2.  This  rule  depends  upon  the  principle  that  a  part  of  a  unit 
may  be  divided  into  other  parts,  as  well  as  a  wJwle  unit. 

2.  A  lad  paid  |f  dollar  for  6  balls:  what  was  that 


Ans.  ^y  dol. 


apiece  ?  Ans.  -fo  d( 

3.  Divide  -fj  by  2.  8.  Divide  if  by  31. 

4.  Divide  fj  by  3.  9.  Divide  -^°/  by  50. 

r»«      •!.    tf    1 /-  _  -  T\  •      •  ,1  -     T  •>  7     1 /T   . 


4.  Divide  ||  by  # 

5.  Divide  -ff  by  6, 

6.  Divide  f^  by  7. 

7.  Divide  Jf  by  11. 


9.   j_/iYiut;  --gy-  uj   ^u. 

10.  Divide  ^j-  by  64. 
11 


Divide  ffj  by  85. 


11.  j^iviut;  -j-j-j  uj 

12.  Divide  f££  by 


100. 


13.  A  drover  paid  18J  dollars  for  5  sheep:  how  much 
was  that  per  head  ? 

Analysis. — Reducing  the  mixed  num-  Operation. 

ber  1 8  i:  to  an  improper  fraction ,  it  becomes  1 8  J — 3J-. 

V  ;  and  ¥+5=ft»  or  3tt  dols.  *£+  5=4$,  or  3JJ. 


118  DIVISION      OF      FRACTIONS. 

14.  If  5  barrels  of  flour  cost  37-J  dollars,  what  will  1 
barrel  cost  ? 

15.  Divide  15J  by  3.  18.  Divide  65^  by  23. 

16.  Divide  22 x  by  6.  19.  Divide  100-ff  by  40. 

17.  Divide  41-J  by  11.  20.  Divide  225^  by  50. 

CASE    II. 
To  Divide  a  Whole  Number  by  a  Fraction. 

1.  At  \  dollar  apiece,  how  many  chickens  can  be 
bought  for  3  dollars  ? 

Analysis.— Since  1  half  dollar  will  buy  1  chicken,  3  dollars 
will  buy  as  many  as  there  are  halves  in  3  dols.,  whicli  are  6. 
Therefore,  3  dols.  will  buy  6  chickens. 

2.  At  I  of  a  dollar  a  quart,  how  many  quarts  of  cher- 
ries can  you  buy  for  5  dollars  ? 

3.  If  you  divide  8  apples  equally  among  4  boys,  what 
part,  and  how  many  will  each  receive  ? 

Analysts. — 1  is  \  of  4 ;  therefore,  each  boy  will  receive  \  part. 
Again,  if  8  apples  are  divided  into  4  equal  parts,  1  part  will  be 
\  of  8,  which  is  2.     Therefore,  etc. 

4.  A  teacher  distributed  16  pounds  of  figs  equally 
among  5  pupils,  what  part,  and  how  many  did  each 
receive  ? 

5.  At  J  of  a  dollar  a  yard,  how  many  yards  of  poplin 
can  be  bought  for  5  dollars? 

Analysis. — In  5  dollars  there  are  15  thirds,  and  2  thirds  are 
contained  in  15  thirds,  7^  times.     Therefore,  etc. 

6.  At  J  of  a  cent  apiece,  how  many  apples  can  I  buy 
for  8  cents  ? 

7.  At  $  of  a  dollar  a  pound,  how  many  pounds  of 
cinnamon  can  I  buy  for  10  dollars  ? 

8.  At  J  of  a  dollar  a  box,  how  many  boxes  of  white 
grapes  can  be  bought  for  6  dollars  ? 


DIVISION      OF      FRACTIONS.  119 

SLATE     EXERCISES. 

i.  At  f  of  a  dollar  a  pound,  how  many  pounds  of 
tea  can  I  bay  for  20  dollars  ? 

Analysis. — At  1  third  dollar  a  Operation. 

pound,  I  can  buy  as  many  pounds        2od.-7-£=:(20  x  3)  — 2. 
as  there  are  thirds  in  20  dollars,  or     (20  x  o\ _^_  2  __6 0    or  IQ  „ 
So  pounds.     But    the  price  is  2        ~  • '  T* 

thirds  dollar  a  pound;  therefore,        v/i,  -so  x  2—  2  —  30  p. 
I  can  buy  only  1  half  of  60  or  30  pounds. 

In  the  operation  we  multiply  the  whole  number  20  by  the 
denominator  3,  and  divide  the  product  by  the  numerator  2.  But 
this  is  the  same  as  inverting  the  fractional  divisor,  and  then 
multiplying  the  dividend  by  it.    (P.  113,  Q.  40.) 

44.  How  divide  a  whole  number  by  a  fraction? 

Multiply  the  whole  number  by  the  fraction  inverted. 

Notes — 1.  The  reason  of  the  rule  is  this:  multiplying  the 
whole  number  by  the  given  denominator  reduces  it  to  a  fraction 
having  the  same  denominator  as  the  given  fraction.  Hence,  the 
numerators  are  like  numbers,  and  one  may  be  divided  by  the  other, 
is  whole  numbers.    (P.  104,  Rem.) 

2.  To  divide  a  whole  by  a  mixed  number,  reduce  the  mixed 
number  to  an  improper  fraction.    (Ex.  2.) 

3.  A  fraction  is  inverted  when  its  terms  are  made  to  exchange 
places.    Thus,  ~  inverted  becomes  |. 

2.  At  1 2\  dollars  apiece,  how  many  ploughs  can  a  man 
buy  for  75  dollars?  Ans.  6. 

3.  Divide  40  by  f.  7.  Divide  96  by  i8|. 

4.  Divide  55  by  f.  8.  Divide  100  by  2cf. 

5.  Divide  68  by  ■&.  9.  Divide  250  by  37-J. 

6.  Divide  75  by  fa  10.  Divide  560  by  6  6  J. 

11.  A  lady  paid  62  dollars  for  15 -J  yards  of  silk:  what 
was  the  silk  a  yard  ? 

12.  If  a  horse  travels  75  miles  in  i8|  hours,  how  far 
will  he  go  in  1  hour  ? 


120  DIVISION      OF      FRACTIONS. 

CASE     III. 

To  Divide  a  Fraction  by  a  Fraction  when  they  have  a 

Common  Denominator. 

i.  At  |  of  a  dollar  a  pound,  how  many  pounds  of 
pepper  can  be  boug  i'~  for  -J-  of  a  dollar  ? 

Analysis. — If  2  thirds  of  a  dollar  will  buy  1  pound,  7  thirds 
will  buy  as  many  pounds  as  2  is  contained  times  in  7,  and  2  is 
contained  in  7,  3^  times.     Therefore,  3  dollar  will  buy  3J  pounds. 

2.  How  many  needles,  at  f  cent  apiece,  can  you  buy 
for  I  cent  ? 

3.  How  many  pen-knives,  at  J  of  a  dollar,  can  be  had 
for  J32-  of  a  dollar  ? 

4.  At  J  of  a  dollar  a  yard,  how  many  yards  of  ribbon 
can  be  purchased  for  £  of  a  dollar  ? 

.  5.  At  I  of  a  dollar  a  yard,  how  many  yards  of  ging- 
ham will  -^  of  a  dollar  buy  ? 

SLATE     EXERCISES. 

Remark. — When  fractions  have  a  common  denominator,  their 
numerators  are  like  numbers.  Hence,  one  numerator  may  bo 
divided  by  the  other,  as  whole  numbers. 

1.  If  f  of  a  dollar  will  buy  1  pound  of  coffee,  how 
many  pounds  can  be  bought  for  -^  of  a  dollar  ? 

Analysis. — If  &  dollar  will   buy  Operation. 

1  pound,  35X  dollar  will  buy  as  many     2  7_^2_2^^2_       » 
jpounds  as  f-  are  contained  times  in 
V,  which  is  134  times.     Therefore,  etc. 

45.  How  divide  one  fraction  by  another  when  they  have  a 
common  denominator? 

Divide  the  numerator  of  the  dividend  by  that  of  the 
divisor. 

2.  Divide  f&  by  ^.  5.  Divide  §£  by  J£. 

3.  Divide  ft  by  •&.  6.  Divide  f|  by  £f. 

4.  Divide  f f  by  ^.  7.  Divide  JJ  by  ft 


DIVISION     OF     FRACTIONS.  121 

To  Divide  a  Fraction  by  a  Fraction,  when  they  have 
Different  Denominators. 

i.  At  -J  of  a  dime  apiece,  how  many  pears  can  be  pur- 
chased  for  j  of  a  dime  ? 

Analysis. — \  dime  will  buy  as  many  opebation. 

pears  as  \  dime  is  contained  times  in  f  1X4 4 

dime.     Reducing  \  and  f  to  a  common  3X5  ' 

denominator,  they  become    jV  and  -,22-.  ,  x  ~ 

Their  numerators  are  now  like  numbers,  =t9;?  > 

and  one  may  be  divided  by  the  other.  7     4 

Thus,  9-5-4=2!  pears,  the  answer  re-       T2~T2  — 9~4 > 

quired.     (P.  120,  Eem.)  9-^4  =  2iP- 

By  inspecting  the  operation  of  redu-  Or,  |xf=|,  or  2^. 
cing  the  fractions  to  a  common  denomi- 
nator, it  will  be  seen  that  the  numerator  of  each  is  multiplied 
into  the  denominator  of  the  other.  This  produces  the  same  com- 
bination of  terms  and  the  same  results  as  inverting  the  divisor 
and  multiplying  the  terms  of  the  dividend  by  it.  Thus,  |-*ii= 
£  x  $=$,  or  z\  pears,  the  same  as  before. 

Remark. — In  dividing,  no  use  is  made  of  the  common  denomi- 
nator ;  hence,  in  practice,  multiplying  the  denominators  together 
may  be  omitted. 

46.  How  divide  a  fraction  by  a  fraction,  when  they  have 
different  denominators? 

Reduce  the  fractions  to  a  com.  denominator,  and  di- 
vide the  numerator  of  the  dividend  by  that  of  the  divisor. 

Or,  multiply  the  dividend  by  the  divisor  inverted. 
Note. — Mixed  numbers  must  be  reduced  to  improper  fraction^ 
and  compound  fractions  to  simple  ones.    (Ex.  17,  24.) 

2.  At  f  of  a  dollar  a  pound,  how  much  tea  can  be 
had  for  f  of  a  dollar  ? 

3.  How  many  pineapples,  at  -^  of  a  dollar  each,  can 
be  had  for  f  of  a  dollar  ? 

4.  At  |  of  a  dollar  a  pound,  how  much  sugar  can  be 
had  for  f  of  a  dollar  ? 

6 


122  DIVISION      OF      FRACTIONS. 

Perform  fche  following  divisions : 

5.  Divide  §■  by  J.  11.  Divide  x|  by  T7T. 

6.  Divide  f  by  f.  12.  Divide  fj  by  £$. 

7.  Divide  f  by  f.  13.  Divide  Jf  by  ^-. 

8.  Divide  T7<j  by  TV  14.  Divide  ^  by  TVo- 

9.  Divide  T\  by  T\.  15.  Divide  Jff  by  ^ 
10.  Divide  ^  by  ^.  16.  Divide  -Jf  f  by  ^f. 

17.  How  many  bushels  of  apples,  at   2  J  dollars  a 
bushel  can  be  purchased  with  8f  dollars  ? 

Analysis. — Reducing  the  Operation. 

mixed  numbers  to  improper  24=JJt,  and  84=^; 

fractions,  we  have  2|=14i>  85  44_^._ii_-44x   4  . 

=  16i«     Inverting  the  divisor,  ^ 
we  cancel  the  common  factor     ^4-xT4T-     —  x  — =1$-,  or  3^. 

11,  and    proceed  as  before.  5       ** 
Alia.  2>\  bushels. 

18.  Divide  3!  by  2J.  21.  Divide  i8|  by  5 -J. 

19.  Divide  8f  by  3 J.  22.  Divide  27  J  by  n|. 

20.  Divide  13  J  by  5^  23.  Divide  55^  by  2  if 

24.  What  is  the  quotient  of  -J  of  f  of  3-J  divided  by 

fof  J? 

Analysis. — Reducing  2>\  to  ¥i  an(*  inverting  the  divisor,  we 
have  \  of  J  of  -L/-*-J  of  !=:$  xfx^xf  x  |=|,  or  ij,  ^Ins. 

25.  Divide  f  of  J  by  r  of  §, 

26.  Divide  f  of  f  of  |-  by  $  of  |> 

27.  Divide  |  of  10 \  by  -J  of  J  of  f. 

28.  Divide  \  of  -J-  of  £  by  |  of  6  J. 

To  reduce  a  Complex  Fraction  to  a  Simple  one, 

21 
1.  Reduce  the  complex  fraction  -3  to  a  simple  one. 

Analysis. — The  given  complex  fraction  1st  Opebation. 

is  equivalent  to  2^-4-4.     Reducing  the  nu-  %\ # 

mcrator  2\  to  a  simple  fraction,  it  becomes  4         T  •  4  > 

5,  and  3-*-4=&>  the  answer  required.     (P.  2£— 4.=X— 4  ■ 


DIVISION      OF      FRACTIONS.  123 

Or,  reducing  both  the  numerator  and  2d  Operation. 

denominator  to  a  simple  fraction,  they  be-         2  j-r4— f-i-f  5 

come  I  and  ±     Now,  $-*-f=*$  x  £=-&,  the  7_r_4_7vi—  ? 

,    ,.  T  •  T — T  A  ?  —  XT' 
same  as  before. 

I 

2.  Reduce  the  complex  fraction  -  to  a  simple  one. 
Solution. — Performing  the  division  indicated,  ^-r-3=£,  Ans. 

47.  How  reduce  a  complex  fraction  to  a  simple  one? 
Reduce  the  numerator  to  a  simple  fraction,  and  divide 

it  by  the  denominator.     (P.  1 1 7,  Q.  43.) 

Remarks. — 1.  Complex  Fractions  when  reduced  to  simple  ones, 
are  added,  subtracted,  etc.,  like  other  Simple  Fractions. 

2.  The  expressions  -~,  -^,  etc.,  indicate  a  division  of  one  frac- 

tional  number  by  another. 

Such  expressions  are  reduced  to  simple  fractions  in  the  samo 
manner  as  one  fraction  is  divided  by  another.     (Ex.  3.) 

2* 

3.  Seduce  the  expression  -|  to  a  simple  fraction. 

5  3 

ANALYSIS. — The  given  expression  is  Operation. 

equivalent  to  2J-4-53-.     Reducing  the  di-  2~2__.    1  .     2  . 

visor  and  dividend  to  simple  fractions,  5 J        2  •  53? 

they  become  $  and  ^  ;  and  1+^—1 x  2}=$,  and  5j=^; 

***     (P.  99,  Q- 250  |H.¥=|xA^ii 

Reduce  the  following  complex  fractions  to  simple  ones, 

4*  6.M.  8.f.  10.£* 

8  6  9  10 

5.  — .  7.  — .  9.  -— .  11.  —^5 

7  20  35  42 

48.  The  preceding  principles  may  be  reduced  to  the 
following 

GENERAL   RULE. 

Reduce  whole  and  mixed  numbers  to  improper  frac- 
tions, compound  and  complex  fractions  to  simple  ones, 
and  multiply  the  dividend  by  the  divisor  inverted. 


124  QUESTIONS      FOR      REVIEW. 

QUESTIONS    FOR    REVIEW. 

i.  If  you  pay  3  J  dollars  a  week  for  board,  what  will 
it  cost  you  to  board  1 1  weeks  ? 

2.  If  a  ton  of  hay  is  worth  17  dollars,  what  is  £  of  a 
ton  worth  ? 

3.  What  will  I  of  a  yard  of  ribbon  cost,  at  f  of  a  dol- 
lar a  yard  ? 

4.  What  cost  10J  pounds  of  butter,  at  \  of  a  dollar  a 
pound  ? 

5.  What  cost  16 J  yards  of  silk,  at  2%  dollars  per  yard? 

6.  At  f  of  a  dollar  a  pound,  how  many  pounds  of  tea 
can  be  purchased  for  30  dollars  ? 

7.  How  many  pen-knives  can  I  buy  for  60  dollars,  if 
I  pay  £  of  a  dollar  apiece  ? 

8.  At  f  of  a  dollar  a  pound,  how  many  pounds  of 
almonds  can  be  bought  for  58!  dollars  ? 

9.  How  much  maple  sugar,  at  \  of  a  dollar  a  pound, 
can  be  purchased  for  ^  of  a  dollar  ? 

10.  At  2 J  dollars  a  cord,  how  much  wood  can  be  had 
for  18  dollars? 

1 1.  How  much  flour,  at  7}  dollars  a  barrel,  can  be  had 
for  37  J  dollars  ? 

1 2.  Required  the  sum  of  J  and  TV  Their  difference. 
Their  product.  The  quotient  of  the  former  divided  by 
the  latter. 

13.  How  many  days  can  you  hire  a  laborer  for  37] 
dollars,  if  you  pay  him  1  j  dollar  a  day  ? 

14.  A  planter  raised  60  bales  of  cotton,  sold  \  of  them 
to  one  merchant,  and  f  to  another:  how  many  bales 
had  he  left? 

15.  A  speculator  bought  a  quantity  of  apples  for  162J 
dollars,  and  sold  them  for  210J  dollars:  what  was  his 
profit  ? 


QUESTIONS      FOE      REVIEW.  125 

1 6.  Bought  15  pounds  of  butter,  at  £  dol.  a  pound; 
and  10  gal.  of  molasses,  at  f  dol.  a  gal.:  what  was  the 
cost  of  both  ? 

17.  At  }  of  a  dollar  a  pound,  how  many  raisins  can 
be  bought  for  ^  of  a  dollar  ? 

18.  What  is  the  quotient  of  ^  of  f  of  5^  divided  by 
fof  J? 

19.  If  I  pay  \  of  f  of  20  dollars  for  a  ton  of  coal,  what 
must  I  pay  for  \  of  4f  tons  ? 

20.  A  man  having  500  dollars,  laid  out  \  of  it  in  cot- 
ton, which  was  \  of  £  of  a  dollar  a  pound :  how  much 
cotton  did  he  have  ? 

21.  A  man  owniug  -fj  of  a  ship,  sold  f  of  his  share  of 
her :  what  part  of  the  ship  did  he  sell,  and  what  part 
had  he  left  ? 

22.  How  long  will  150  pounds  of  coffee  last  a  family, 
if  they  use  3 \  pounds  a  week  ? 

23.  A  and  B  drew  a  prize  amounting  to  256^  dollars; 
A  took  i6of  dollars :  how  much  did  B  have  ? 

24.  What  will  it  cost  to  build  -J  of  f  of  16J  rods  of 
stone  wrall,  at  if  of  a  dollar  a  rod  ? 

25.  If  2\  of  a  yard  of  velvet  can  be  bought  for  12 
dollars,  what  part  of  a  yard  can  be  bought  for  1  dollar  ? 

26.  Divide  f  of  J  by  f  of  f 

27.  Divide  f  of  32  by  §  of  f. 

28.  Divide  -fc  of  i6£  by  f  of  10. 

29.  Divide  J  of  f  of  18J  by  f  of  3$. 

30.  Which  will  cost  more,  8  barrels  of  flour,  at  7^ 
dollars  a  barrel,  or  16  barrels  of  potatoes,  at  3 \  dollars  a 
barrel ? 

31.  If  oranges  are  6\  cents  apiece,  how  many  can  be 
bought  for  87  J  cents  ? 

32.  At  i6f  cents  a  pound,  how  much  lard  can  be 
bought  for  83  J  cents  ? 


126  FRACTIONAL     RELATION 

FRACTIONAL    RELATION     OF    NUMBERS. 

To  find  what  part  one  number  is  of  another. 

Remark. — That  numbers  may  be  compared  with  each  other. 
they  must  be  so  far  of  the  same  nature  that  one  may  properly  be 
said  to  be  apart  of  the  other.  Thus,  a.  foot  may  be  compared  with 
a  yard;  for,  one  is  a  third  part  of  the  other.  But  it  can  not  be 
said  that  afoot  is  a  part  of  a  pound;  therefore  the  former  can  nc  * 
be  compared  with  the  latter. 

i.  What  part  of  3  is  i  ? 

Analysis. — If  3  is  divided  in  3  equal  parts,  one  of  these  parts 
is  1  third.     Therefore,  1  is  ^  part  of  3. 

2.  What  part  of  3  is  2  ? 

Analysis. — 2  is  2  times  1  third  part  of  3,  or  2  thirds  of  3. 

3.  In  1  gallon  there  are  4  quarts :  what  part  of  a  gal- 
lon is  1  quart  ?     What  part  is  3  quarts  ? 

4.  What  part  of  5  is  3  ?    Is  4  ?    Is  2  ?    Is  1  ? 

5.  What  part  of  6  is  2  ?    Is  3  ?    Is  4?    Is  5  ? 

6.  What  part  of  4  apples  are  5  apples  ? 

Analysis. — 1  apple  is  1  fourth  part  of  4  apples;  therefore,  5 
apples  must  be  5  times  1  fourth,  or  5  fourths  of  4  apples. 

7.  What  part  of  8  pounds  is  9  pounds  ?  Is  1 1  pounds  ? 

SLATE     EXERCISES. 

1.  What  part  of  5  cents  is  3  cents  ? 
Analysis. — 3  cents  are  3  times  £,  or  |  of  5  cents. 
49.  How  find  what  part  one  number  is  of  another  J 

Make  the  number  denoting  the  part  the  numerator, 
and  that  with  which  it  is  compared  the  denominator. 

Note. — The  fraction  thus  found  should  be  reduced  to  its 
lowest  terms. 

2.  What  part  of  48  is  1 2  ?     Of  63  is  28  ? 

3.  What  part  of  81  is  27  ?    Of  90  is  63  ?    Of  100  is  40? 


OF     NUMBERS.  127 

4.  What  part  of  35  dollars  is  19  dollars  ? 

5.  If  I  divide  a  bushel  of  plums  equally  among  15 
boys,  what  part  of  a  bushel  will  1  boy  receive  ?  What 
part  will  9  boys  receive  ? 

6.  Helen's  age  is  18  years,  and  her  brother's  14:  what 
part  of  her  age  is  her  brother's  ? 

7.  Henry  has  91  marbles,  and  Charles  70:  Charles' 
marbles  are  equal  to  what  part  of  Henry's  ? 

8.  If  5  pencils  cost  17  cents,  what  will  4  pencils  cost? 

Analysis. — 4  pencils  are    f  of    5   pencils  ;  5)17    cts. 

therefore,  4  pencils  will  cost   f   of   17  cents.  

Now,  i  of  17  cents  is  3^  cents,  and  4  fifths  are  3y 

4  times  3!  cents,  or   13!   cents,  the  cost  re-  4 

quired.  .  7    , 

^  Am.  13  J  cts. 

9.  If  8  oranges  cost  32  cents,  what  will  6  cost  ? 

10.  If  20  cows  cost  625  dollars,  what  will  35  cost? 

11.  If  13  sofas  cost  572  dollars,  what  will  6  cost  ? 

12.  What  part  of  4  pears  is  f  of  a  pear  ? 

Analysis. — 1  pear  is  i  part  of  4  pears  ;  Operation. 

hence,  f  of  a  pear  must  be  f  of  $=-ft-,  or  £  2 

of  a  pear.     Therefore,  etc.  7     T~4? 

Making  the  fraction  which  denotes   the  2        _  2        .1 

part  the  numerator,  and  the  tcfo>&  number  the  T  •  4  — T2">       3"* 

3. 

denominator,  we  have  the  complex  fraction  -,  to  be  reduced  to  a 

simple  one.     (P.  123,  Q.  47.) 

Or,  what  is  the  same  thing,  a  fraction  to  he  divided  by  a  whole 
lumber  ;  and  f-M=A,  or  |.    (P.  117,  Q.  43.) 

13.  What  part  of  15  is  f  ?     15.  What  part  of  45  is  -ft-? 

14.  What  part  of  26  is  £  ?     16.  What  part  of  63  is  -ft  ? 

17.  If  15  barrels  of  flour  cost  100  dollars,  what  will  J 
of  a  barrel  come  to  ? 

18.  If  20  acres  of  land  jMd  250  bushels  of  corn,  what 
will  I  of  an  acre  yield  ? 


128  FRACTIONAL     RELATION 

To  find  a  number  when  a  part  of  it  is  given. 

i.  5  is  |  of  what  number? 

Analysis. — If  5  is  £,  3  thirds,  or  the  whole  number,  must  be  3 
times  5,  or  15.     Therefore,  5  is  £  of  15. 

2.  6  is  \  of  what  number  ?     7  is  £  of  what  number  ? 

3.  8  is  f  of  what  number  ? 

Analysis. — Since  8  is  f  of  a  certain  number,  1  third  is  £  of  8„ 
which  is  4,  and  3  thirds  must  be  3  times  4,  or  12.     Therefore,  etc. 

5.  George  has  12  apples,  which  are  f  of  the  number 
which  William  has :  how  many  has  William  ? 


SLATE    EXERCISES. 

I 


1.  16  is  f  of  what  number? 


1st  Analysis. — Since  f  of  a  number  1st  Operation. 

is  16,  $  or  the  whole  number,  must  be  as        i6-t-t=i6x4; 

many  units  as  |  is  contained  times  in  16;  ^      3_  48 

and  i6-s-f=i6  x  f ,  or  24.  ?     ~Z  > 

2d  Analysis. — Since  16  is  §  of  a  certain  .    _ 

,.,._,                ,                  ,      ,             2d  Operatiow. 
number,  1  third  of  that  number  must  be  ■$•         T  _    ,.        ~ 

of  16,  which  is  8.    3  thirds,  or  the  whole        ^~ 

number  must  be  3  times  8,  or  24.  "3"— °  x  3  —  24« 

50.  How  find  a  number  when  a  part  of  it  is  given  ? 
Divide  the  number  denoting  the  part  by  the  fraction. 
Or,  find  one  part  as  indicated  by  the.  numerator  of  the 
fraction,  and  multiply  this  by  the  denominator. 

2.  32  is  \  of  what?  5.  100  is  §  of  what? 

3.  45  is  -f  of  what?  6.  144  is  \%  of  what? 

4.  72  is  f  of  what  ?  7.  250  is  \\  of  what  ? 

8.  If  -J  of  an  acre  of  land  is  worth  35  dollars,  what  is 
a  whole  acre  worth  ? 

9.  A  man  paid  75  dollars  toward  a  horse,  which  was 
T7T  of  the  price:  what  did  he  give  for  the  horse? 

10.  A  man  being  asked  how  old  he  was,  replied  that 
r72  of  his  age  equaled  49  years :  what  was  his  age  ? 


DECIMAL    FRACTIONS 


PRELIMINARY     EXERCISES. 

1 .  If  a  sheet  of  paper  is  divided  into  10  equal  parts,  what  part 
of  a  sheet  is  i  of  these  parts  ? 

One  of  these  parts  is  TV  of  a  sheet. 

2.  If  one  of  these  tenths  is  divided  into  10  other  equal  parts, 
what  part  of  a  sheet  is  i  of  these  parts  1 

One  of  these  parts  is  -^  of  ^,  or  y-J ^  of  a  sheet. 

3.  If  one  of  these  hundredths  is  divided  into  io  other  equa^ 
parts,  what  part  of  a  sheet  is  i  of  these  parts  ? 

Each  part  is  -^  of  ^  of  -fa,  or  yoVtf  0I>  a  sheet 

4.  What  is  meant  by  a  tenth,  a  hundredth,  a  thousandth,  etc.  V 
A  tenth  is  one  of  the  ten  equal  parts  into  which  a 

number  or  thing  may  be  divided,  etc.  ? 

5.  How  much  greater  are  tens  than  units;  hundreds  than 
tens ;  thousands  than  hundreds,  etc.  ? 

Tens  are  io  times  greater  than  units,  and  each  suc- 
ceeding order  is  io  times  greater  than  the  preceding. 

6.  How  much  less  are  tenths  than  units;  hundredths  than 
tenths  ;  thousandths  than  hundredths,  etc.  ? 

Tenths  are  io  times  less  than  units  ;  hundredths  are 
io  times  less  than  tenths;  and  so  on,  each  succeeding 
order  being  io  times  less  than  the  preceding. 

7.  What  places  do  tens,  hundreds,  thousands,  etc.,  occupy  ? 
Tens  occupy  the  first  place  on  the  left  of  units ;  hun- 
dreds, the  second  ;  thousands,  the  third,  etc. 

8.  Following  this  analogy,  what  place  should  tenths,  hun- 
dredths, thousandths,  etc.,  occupy  ? 

Tenths  in  the  decreasing  scale  correspond  with  tens 
in  the  increasing  scale ;  hence  they  should  occupy  the 
first  place  on  the  right  of  units.  In  like  manner,  hun- 
dredths, which  correspond  with  hundreds,  should  occupy 
the  second  place  ;  thousandths,  the,  third  place,  etc. 


130  NOTATION      OF      DECIMALS. 

9.  How  many  units  make  a  ten,  tens  a  hundred,  etc.  ? 

10.  How  do  the  orders  of  whole  numbers  increase  ? 
They  increase  from  right  to  left  by  the  scale  of  io. 

11.  How  many  tenths  make  a  unit;  hundredths  a  tenth: 
thousandths  a  hundredth,  etc. 

Ten  -  tenths  make  a  unit ;  ten  hundredths  make  a 
tenth  ;  ten  thousandths  make  a  hundredth,  etc. 

12.  How  do  the  orders  of  these  fractions  decrease  ? 
They  decrease  from  left  to  right  by  the  scale  of  io. 

NOTATION     OF     DECIMALS. 

13.  What  are  Decimal  Fractions  ? 

Decimal  Fractions  are  those  in  which  the  unit 
is  divided  into  tenths,  hundredths,  thousandths,  etc. 

They  arise  from  dividing  a  unit  into  ten  equal  parts, 
or  tenths ;  then  subdividing  one  of  these  tenths  into 
ten  other  equal  parts,  or  hundredths ;  and  so  on,  the 
successive  orders  decreasing  regularly  by  the  scale  of  io. 

14.  How,  and  upon  what  principle  are  they  expressed  ? 

By  placing  a  point  before  the  numerator,  and  assign- 
ing to  each  figure  a  value  according  to  the  place  it  occu- 
pies, as  in  whole  numbers.  Thus,  -fa  is  expressed  by 
writing  3  in  the  first  place  on  the  right  of  units ;  as,  .3 ; 
■jjj-Q  by  writing  3  in  the  second  place ;  as,  .03  ;  t^  by 
writing  3  in  the  third  place ;  as,  .003. 

15.  What  do  figures  standing  in  the  first,  second,  third,  etc., 
places  on  the  right  of  units  denote  ? 

When  standing  in  the  first  place  on  the  right  of  uni 
they  denote  tenths;  in  the  second  place,  they  denote 
hundredths  ;  in  the  third  place,  thousandths,  etc. 

Notes. — 1.  The  point  used  to  distinguish  decimals  from  ibholt 
number*,  is  called  the  decimal  'point. 

2.  'i'heso  fractions  are  called  decimals  from  the  Latin  decern, 
ten,  which  indicates  their  origin  and  stale  of  decrease. 


NOTATION      OF      DECIMALS.  13 1 

16.  What  is  the  denominator  of  a  decimal  fraction  ? 
It  is  always  10,  ioo,  iooo,  etc.,  or  i  with  as  many 
ciphers  annexed  as  there  are  decimals  in  the  numerator. 
Name  the  orders  of  integers,  beginning  at  units. 
Name  the  orders  "of  decimals,  beginning  at  units  place. 

TABLE. 


1 

/" N 

CO 
M 

■a 

S3 

2 

o- 

CO 

p 

1 

•g 

OQ 

I 

1 

00 

"8 

s 

1 

CO 

1 

O 

3 

00 

OS* 

1 

1 

| 

S 

P 

"3 

02 

s 

I 

1 

03 

B 

CD 

i 

-3 

1 

CD 

QQ* 
i—i 

p 
s 

B 

1 

P 

5 
p 

1 

g 

p 

i 

6 

5 

2 

3 

4 

7 

3 

• 

5 

2 

8 

7 

3 

5 

In 

itegers. 

v  

Decimals, 

1 7.  What  is  the  effect  of  prefixing  ciphers  to  decimals  ? 
Each   cipher  prefixed  to  a  decimal,  diminishes  its 

value  ten  times,  or  divides  it  by  io. 

18.  What  is  the  effect  of  annexing  ciphers  to  decimals? 
The  value  is  not  altered.     Thus,  .3 =.30 =.300,  etc. 

19.  How  write  decimals  ? 

Write  the  figures  of  the  numerator  in  their  order,  as- 
signing to  each  its  proper  place  Mow  units,  and  prefix  to 
them  the  decimal  point. 

If  the  numerator  has  not- as  many  figures  as  required, 
supply  the  deficiency  oy  prefixing  ciphers. 

Note. — A  decimal  and  integer  written  together,  are  called  a 
mixed  number  ;  as,  35.263.    (P.  92,  Q.  13.) 

i.  On  which  side  of  units  are  tens  ?  Tenths  ?  Thou- 
sands ?    Hundredths  ?    Hundreds  ?     Thousandths  ? 

2.  What  is  the  name  of  the  second  place  on  the  right 
of  units  ?     The  fourth  ?     The  third  ?     The  fifth  ? 


132  NOTATION      OF      DECIMALS. 

3.  How  many  decimal  places  are  required  to  express 
tenths  ?     Thousandths  ?    Hundredths  ?    Millionths  ? 


SLATE      EXERCISES. 

Write  the  following  fractions  decimally  : 

1.  \z  hundredths.  7.  9  thousandths. 

2.  25  hundredths.  8.  13  thousandths. 

3.  5  hundredths.  9.  TWA- 

4.  49  hundredths.  10.  yf^. 

5.  119  thousandths.  11.  ttrrttt* 

6.  27  thousandths.  12.  x^^. 

13.  Write  6  hundredths.     41  thousandths.     7  thou- 
sandths. 

14.  Write  201  ten-thousandths.     752  hundred-thou- 
sandths. 

15.  Write  5   millionths.     63  millionths.     98  mil- 
lionths.    375  millionths. 

20.  How  read  decimals  ? 

Bead  the  decimals  as  whole  numbers,  and  apply  to 
them  the  name  of  the  lowest  order. 

Remark. — The  unit's  place  should  always  be  the  starting  point 
both  in  reading  and  writing  decimals. 

Copy  and  read  the  following : 

15.  .7.  21.  2.35.  27.  21.251.  S3-  I2I.4502- 

16.  .75.  22.  3.236.  28.  30.4312.  34.  240.4023.  t 

17.  .06.  23.  5.078.  29.  44.0643.'  35.  306.46531. 

18.  .121.  24.  6.2356.  30.  53.21034.  36.  500.00729. 

19.  .065.  25.  7.3062.  31.  72.05213.  37.  607.329267. 

20.  .008.  26.  8.5602.  32.  84.00605.  38.  730.004308. 

***  Dictation  exercises  in  reading  and  writing  decimals  should 
be  practiced  till  the  class  is  perfectly  familiar  with  them. 


REDUCTION      OF      DECIMALS.  133 

REDUCTION     OF     DECIMALS. 

CASE     I. 

To  Reduce  Decimals  to  Common  Fractions. 

i.  Reduce  .27  to  a  common  fraction. 

Analysis. — Since  .27  has  two  decimal  fig-  Operation. 

tires,  its  denominator  must  be  100.    Hence,       '^—ToV?  -4w& 
aj=-fifa.    In  the  operation  we  omit  the  deci- 
mal point,  and  place  the  denominator  100  under  the  27. 

91.  How  reduce  decimals  to  common  fractions? 
Erase  the  decimal  point,  and  place  the  denominator 
under  the  numerator,    (P.  131,  Q.  16.) 

Note. — After  decimals  are  reduced  to  common  fractions,  they 
may  be  reduced  to  lower  terms,  to  a  common  denominator,  etc., 
and  then  be  treated  in  all  respects  like  other  common  fractions. 

2.  Reduce  .35  to  a  common  fraction,  and  to  its  lowest 
terms.  Am.  .35  =fw>  and  TVo  =  27o- 

Reduce .  the  following  decimals  to  common  fractions 
in  their  lowest  terms : 

3.  .24.  7.  .04.  11.  .4032.  15.  .00045. 

4.  .135.         8.  .025.  12.  .0005.  16.  .00328. 

5.  .404.         9.  .204.  13.  .0106.  17.  .01032. 

6.  .675.       10.  .1025.         14.  .7524.  18.  .123456. 

CASE    II. 
To  Reduce  Common  Fractions  to  Decimals. 
1.  Reduce  J  to  a  decimal. 

Analysis. — \  is  equivalent  to  1  divided  by  4.         Operation. 
But  1  cannot  be  divided  by  4  ;  we  therefore  reduce  4 )  1.00 

it  to  tenths  by  annexing  a  cipher  to  it,  making  10  ~^Z 

tenths.    (P.  57,  Q.  18.)    Now  \  of  10  tenths=2 
tenths  and  2  tenths  over.     Reducing  the  2  tenths  to  hundredths 
by  annexing  a  cipher,  we  have  20  hundredths;  and  £  of  20 
hundred ths= 5  hundredths.     Therefore,  \  equals  .25. 


134  REDUCTION      OF      DECIMALS. 

22.  How  reduce  common  fractions  to  decimals  ? 

Annex  ciphers  to  the  numerator,  and  divide  by  tlie  de- 
nominator. 

Finally,  point  off  as  many  decimal  figures  in  the  result 
as  there  are  ciphers  annexed  to  the  numerator. 

Reduce  the  following  fractions  to  decimals : 

2.  \.  6.  £.  io.  -ft-  1 4.  ^. 

3-  I-  7-  I-      .  ii.  U-  i5-  Tib- 

4-  I-  a  A-  12.  TV  1 6.  j&. 

5-  £•  9-  tV  13-  &  17-  iU> 

18.  Keduce  J  to  the  form  of  a  decimal. 
Analysis. — Annexing  ciphers  to  the  nu-  Operation. 

merator,  and  dividing  by  the  denominator  as      3  )  I'.oooo 
before,  the  quotient  is  3  repeated  continually,  ^VVV3   etc. 

and  the  remainder  is  always  1.     Hence,  &  can- 
not be  exactly  expressed  by  decimals. 

1 9.  Reduce  J  J  to  the  form  of  a  decimal. 

Analysis.-— Annexing  ciphers  and  divid-  Operation. 

ing  as  before,  the  quotient  is  45  repeated      33  )  15.0000 
continually,  and  the  remainder  is  alternate-  4545    etc. 

ly  18  and  15,  the  latter  being  the  given  nu- 
merator.    Therefore,  ^§  cannot  be  exactly  expressed  by  decimals. 

23.  When  the  numerator  with  ciphers  annexed  is  exactly 
iivisible  by  the  denominator,  what  is  the  decimal  called  ? 

It  is  called  a  Terminate  decimal. 

24.  When  it  is  not  exactly  divisible,  and  the  same  figure  or 
set  of  figures  continually  recurs  in  the  quotient,  what  is  the 
decimal  called  ? 

It  is  called  an  Inlerminate,  or  Circulating  decimal. 

25.  What  are  the  figure  or  figures  repeated  called  ? 
They  are  called  the  Repetend. 

Note. — When  the  quotient  has  been  carried  as  far  as  desirable, 
fhe  sign  ( 4-)  is  annexed  to  it,  to  indicate  there  is  still  a  remainder. 


A  D  D I  T  I-ON    OP    DECIMALS.  135 

ADDITION    OF    DECIMALS. 

i.  If  an  arithmetic  costs  6  tenths  dollar,  and  a  gram- 
liar  8  tenths  dollar,  what  will  both  cost  ? 

Analysis. — Since  each  of  these  decimals  expresses  tentlis 
they  have  a  common  denominator,  viz.,  10 ;  therefore,  they 
are  like  numbers,  and  may  be  added,  as  whole  numbers.  Now  £ 
tenths  and  8  tenths  are  14  tenths ;  equal  to  1  and  4  tenths  dollar 

25,  a.  When  have  decimals  a  common  denominator  ? 

Decimals  have  a  common  denominator  when  their  nu- 
merators have  the  same  numher  of  decimal  figures.  As 
.05  and  .07,  whose  denominator  is  100.     (P.  131,  Q.  16.) 

25,  b.  How  reduce  decimals  .to  a  common  denominator  ? 

Make  the  number  of  decimal  figures  the  same  in  each,  by 
annexing  ciphers.     (P.  131,  Q.  18.)     Thus,  .3  and  .05  re- 
duced to  a  common  denominator  become  .30  and  .05. 
•  1.  What  is  the  sum  of  42.136 ;  6.35  ;  13.7  ;  and  .245  ? 

Analysis. — Reduce  the   decimals  to  a  com-       Operation. 
mon  denominator  by  annexing  ciphers,  or,  which  42,I3" 

is  the   same,   write  units  under    units,  tenths  °-35° 

under  tenths,  etc.,  the  decimal  points  being  in  a  I3*7°° 

perpendicular  line.     Beginning  at  the  right,  add  *245 

as    in   whole  numbers,  and  place  the  decimal     A?IS.  62.431 
point  in  the  amount  under  those  in  the  numbers 
added.     (P.  25,  Q.  9.) 

2 a.  How  add  decimals? 

I.  Write  the  numbers  so  that  the  decimal  points  shal 
stand  one  under  another,  with  tenths  tinder  tenths,  etc. 

II.  Beginning  at  the  right,  add  as  in  tohole  numbers, 
,  and  place  the  decimal  point  in  the  amount  under  those 

in  the  numbers  added.     (P.  28,  Q.  13.) 

Rem.— Placing  tenths  under  tenths,  hundredths  under  hun- 
dredths, etc.,  in  effect  reduces  decimals  to  a  common  denominator  / 
hence,  the  ciphers  on  the  right  may  be  omitted  in  the  operation. 
(P.  C31,  Q.  18.) 


16 

ADDITION      OF      DECIMALS. 

N 

(30 

(4.) 

(5.) 

26.176 

8.65 

206.451 

3-7056 

2.5 

.372 

40.45 

.045 

4.38 

i.6 

3-6 

.06 

.023 

5405 

23-75 

2.841 

6.  What  is  the  sum  of  seventeen  and  four  tenths;  si.. 
and  two  hundredths ;  eight  and  forty-five  thousandths  ? 

7.  What  is  the  sum  of  13.71  yards;  21.2  yards;  and 
10.75  yards  ? 

8.  How  many  dollars  in  3  purses ;  the  first  contain- 
ing 26.5  dollars ;  the  second,  1 7.25  dollars ;  and  the 
third,  30.625  dollars  ? 

9.  How  many  acres  in  four  lots,  which  respectively 
contain  19.275  acres;  30.41  acres;  23.261  acres;  and 
31.027  acres? 

10.  What  is  the  sum  of  42.07  gallons  +  50.128  g;  Is. 
4-  1.625  gals-  +  l6.oi8  gals.  ? 

11.  What  is  the  sum  of  28.16  rods  +  45.025  rods  f 
85.7  rods  +  17.265  rods. 

SUBTRACTION     OF     DECIMALS. 

1.  A  lad  having  8  tenths  of  a  dollar,  paid  3  tenth,  of 
dollar  for  his  lunch  :  how  much  had  he  left  ? 
Analysis. — 3  tenths  from  8  tenths  leave  5  tenths.  Therefore  itc. 

2.  Take  7  tenths  from  9  tenths. 

3.  What  is  the  difference  between  .19  and  .17  ? 

4.  Paid  .37  dol.  for  a  bushel  of  apples,  and  .60  rtoL ' 
for  a  bushel  of  corn :  what  was  the  difference  in  price  ? 

5.  A  man  having  .75  of  an  acre  of  land,  sold  .48  of  an 
acre:  how  much  land  did  he  have  left? 

6.  What  is  the  difference  between  .93  and  .62  ? 


SUBTRACTION      OF      DECIMALS.         l37 

SLATE     EXERCISES. 

I.  What  is  the  difference  between  2.34  and  .543  ? 
Analysis. — Reduce  the  decimals  to  a  common  de-  Operation. 

nominator  by  annexing  ciphers,  or  by  writing  units-  2«34 

under  units,  tenths  under   tenths,  etc.,  the  decimal  «543 

points  being  in  a  perpendicular  line.      (P.  38,  Q.  9.)  i«797 
Beginning  at  the  right,  we  see  that  3  thousandths 

can  not  be  taken  from  o  thousandths ;  hence  we  borrow  10,  and 
proceed  as  in  whole  numbers. 

27.  How  subtract  decimals  ? 

L  Write  the  less  number  under  the  greater,  so  that  the 
decimal  points  shall  stand  one  under  the  other,  with 
tenths  under  tenths,  etc. 

II.  Beginning  at  the  right,  subtract  as  in  tvhole  num- 
bers, and  place  the  decimal  point  in  the  remainder  under 
that  in  the  subtrahend.    (P.  42,  Q.  15.) 


Ml 

(3.) 

(4.) 

(50 

From 

6.432 

13.206 

28.3607 

1.00042 

Take 

3-i7 

7.0378  • 

.981 

.236 

Perform  the  subtractions  indicated  in  the  following : 

6.  63.025  —  13.5.  11.  60.001  —  45.008. 

7.  7.46  —  3.678.  12.  1.0006  —  0.37. 

8.  100.007  —  0.845.  13.  0.05  —  0.005. 

9.  275  —  60.75.  14.  0.006  —  0.0006. 

10.  17.4  —  10.0008.  15.  0.0001  —  .00001. 

16.  Sold  2  pieces  of  cloth,  one  37.5  yards  long,  the  other 
31  \ yards:  what  was  the  difference  in  their  length  ? 

17.  A  man  owning  7  tenths  of  a  ship,  sold  25  hun- 
dredths of  her :  how  much  had  he  left  ? 

18.  If  from  150.05  acres  of  land,  87-J  acres  arc  taken, 
how  much  will  be  left  ? 


138        MULTIPLICATION      OP      DECIMALS. 

MULTIPLICATION  .OF    DECIMALS. 

i.  At  .5  of  a  cent  apiece,  what  will  7  apples  come  to  ? 

Analysis. — Since  1  apple  costs  5  tenths  of  a  cent,  7  apples  will 
cost  7  times  5  tenths,  or  35  tenths  of  a  cent ;  and  35  tenths  &ro 
equal  to  3.5  cents.     Therefore,  etc. 

2.  What  cost  4  oranges,  at  .6  of  a  dime  apiece  ? 

3.  At  .3  dollar  apiece,  what  will  6  melons  cost  ?  • 

4.  How  many  tenths  are  5  times  7  tenths  ?  6  times 
4  tenths  ? 

5.  How  many  hundredths  are  3  times  .15  ? 

6.  How  many  tenths  in  4  times  .25  ? 

SLATE    EXERCISES. 

1.  If  1  yard  of  muslin  costs  .25  dollar,  what  will  7 
yards  amount  to  ? 

Analysis. — .25  are  equal  to  2  tenths  and  5  hun-  Operation. 

dredths.     Now  7  times  5  hundredths  are  35  hun-  t2K  dol. 

dredths,  equal  to  3  tenths  and  5  hundredths.    Set  - 

the  5  in  hundredths  place,  and  carry  the  3  to  the  pro-      

duct  of  tenths.     7  times  2  tenths  are  14  tenths,  and  3  1.75  dol. 
are  17  tenths,  equal  to  1  unit  and  7  tenths.    Write 
the  7  in  tenths  place,  and  the  1  in  units  place. 

2.  Multiply  .375  dollar  by  5.  Ans.  1.875  dol. 

3.  At  .75  dollar  a  yard,  what  cost  .5  yard  of  delaine? 

ANALYSIS.— .75  =11oAj,   and   .5 =-^r.     (P.  I33>         Operation. 
Q.  21.)     Now  ^  times  -ftfo=-ftftfr= 375 -*■  1000,  -75  dol. 

ar .  3  7  5 ,  A  ns.    Instead  of  multiply  i  ng  -^  by  &>  *5 

in  the  operation  we  multiply  the  decimals  as     Ans.  .375  dol. 
whole  numbers ;  consequently  the  product  is  as 
many  times  too  large  as  there  are  units  in  the  product  of  their 
denominators,  viz.,  1000.     To  correct  this,  we  point  off  3  figures 
on  the  right  of  the  product,  which  divides  it  by  1000. 

By  inspecting  these  operations,  it  will  be  seen  that  each  product 
has  as  many  decimal  figures  as  both  its  factors. 

In  like  manner  it  may  be  shown,  that  the  product  of  any  two 
decimals  must  have  as  many  decimal  figures  as  both  factors. 


MULTIPLICATION      OF      DECIMALS.        139 

28.  How  multiply  decimals? 

Mulliply  as  in  whole  numbers,  and  from  the  right  .of 
the  product,  point  off  as  many  figures  for  decimals  as 
there  are  decimal  jrtaces  in  both  factors. 

Remakes. — i.  If  the  product  has  not  as  many  figures  as  there 
are  decimals  in  both  factors,  the  deficiency  must  be  supplied  by 
prefixing  ciphers.    (Ex  4.) 

2.  To  multiply  a  decimal  by  10,  100,  1000,  etc.,  remove  the  deci- 
mal point  as  many  figures  to  the  right  as  there  are  ciphers  in  the 
multiplier.  For,  each  removal  of  the  decimal  point  one  place  to 
the  right,  multiplies  the  number  by  10. 

4.  What  is  the  product  of  .004  multiplied  by  .03  ? 

Analysis. — The  product  of   the  significant        Operation. 
figures  4  x  3=12.    But  there  are  five  decimals  in  .004 

the  given  factors  ;  therefore  the  product  must 
have  five  decimals.     Prefixing  three  ciphers  to 


•03 


12,  it  becomes  .00012.  .00012  Ans. 

(5-)  (6-)  (7-)                  (8.) 

Mult.     .127  3-°25  .0046             250.07 

By         .03  .012  .23                 3.04 


Perform  the  following  multiplications : 

.9.  8.02x3.2.      13.  38.065  x. 003.       17.  25.012  x  2.15. 

10.  3.51  x. 09.      14.  506.12  x. 016.       18.  ioooox.007. 

11.  9.027x13.     15.  407.01  x.i 23.       19.  000.01x300.1 

12.  365  x  .05.       16.  1.0004  x. 006.      2°«  0.0004x2.01; 

21.  Allowing  5.5  yards  to  a  rod,  how  many  yards  are 
there  in  20.25  rods? 

22.  If  a  man  earns  1.25  dol.  a  day,  how  much  will  he 
earn  in  19.5  days  ? 

23.  How  many  pounds  of  coffee  in  10^  sacks,  allow- 
ing 37-5  pounds  to  a  sack  ? 


140  DIVISION      OF      DECIMALS. 

24.  If  a  gallon  of  molasses  is  worth  .54  dol.,  how 
much  are  18.75  gallons  worth  ? 
.  25.  What  is  the  product  of  1.005  multiplied  by  .008  ? 

26.  What  is  the  product  of  one  thousandth  into  seven 
hundredths  ? 

27.  What  is  the  product  of  five  ten-thousandths  into 
seven  tenths  ? 

DIVISION     OF     DECIMALS. 
MENTAL    EXERCISES. 

1.  At  2  tenths  of  a  dime  apiece,  how  many  oranges 
can*  a  lad  buy  for  8  tenths  of  a  dime  ? 

Analysis. — 2  tenths  dime  are  contained  in  8  tenths  dime,  4 
times.    Therefore,  etc.    (P.  63,  Q.  10.) 

2.  How  many  times  3  tenths  in  6  tenths  of  a  dollar  ? 

3.  How  many  times  are  7  hundredths  contained  in 
35  hundredths  ?     .4  in  .8  ?    yf^  in  -fifa  ? 

4.  If  a  man  pays  6  tenths  of  a  dollar  for  2  tenths  of  a 
barrel  of  apples,  what  must  he  pay  for  1  tenth  of  a  barrel  ? 

Analysis. — The  object  is  to  divide  6  into  2  equal  parts.  (P. 
63,  Q.  10.)  Since  these  fractions  have  a  common  denominator,  one 
numerator  may  be  divided  by  the  other  like  whole  numbers. 

5.  How  many  times  are  .08  contained  in  .64  ? 

6.  Divide  .4  by  .2  ;  .6  by  .3  ;  .8  by  .4. 

7.  Divide  .08  by  .02;  .16  by  .04. 

SLATE    EXERCISES. 

1.  How  many  times  .2  in  .6  ? 

Analysis. — Since  these  decimals  have  a         opekatiow. 
common  denominator,  they  are  like  numbers;  ,2).6 

hence,  one  can  be  divided  by  the  other  as     Ans.   %  times! 
common    fractions,  and    the  quotient   is   a 
whole  number.    (Page  121,  Q.  46.) 


DIVISION     0*     DECIMALS.  14l 

2.  How  many  times  .04  in  .3  ? 

Analysis. — Reducing   the   given  decimals  to  a   operation. 
common  denominator,  we  have  .04  and  .30.    Now,      ,c>4).30 
.04  is  in  .30,  7  times  and  .02  remainder.    We  put  the      Ans    7~T 
7  in  units'  place,  because  it  is  units.    Annexing  a 
cipher  to  the  remainder,  .04  is  in  .020,  .5  times  and  o  remainder. 
We  set  the  5  in  tentM  place.  Ans.  7.5  times. 

3.  How  many  times  .4  in  .012  ? 

Analysis. — Reducing  these  decimals  to  a  com-  operation. 
mon  denominator,  we  have  .4  =  .400  and  .012.  .4oo).oi2 
Now,  as  .400  is  not  contained  in  .012,  we  put  a  a  ZZ    ^7^> 

cipher  in  units'  place.     Annexing  a  cipher  to  the 
dividend,  we  find  .400  is  not  contained  in  .0120 ;  we  therefore  put 
a  cipher  in  tenths' place.    Annexing  another  cipher,  .400  is  in 
.01200,  3  hundredths  time  and  o  remainder.        Ans.  .03  times. 

29.  How  are  decimals  divided? 

Reduce  the  decimals  to  a  common  denominator,  and 
divide  the  numerator  of  the  dividend  by  that  of  the  divi- 
sor, placing  a  decimal  point  on  the  right  of  the  quotient. 

Annex  ciphers  to  the  remainder,  and  divide  as  before. 
The  figures  on  the  left  of  the  decimal  point  denote  whole 
numbers  ;  those  on  the  right,  decimals. . 

Or,  divide  as  in  whole  numbers,  and  from  the  right 
of  the  quotient,  point  off  as  many  decimals  as  the  deci- 
mal places  in  the  dividend  exceed  those  in  the  divisor. 

Remarks. — 1.  If  there  are  not  figures  enough  in  the  quotient 
for  the  decimals  required  by  the  second  method,  prefix  ciphers. 

2.  To  divide  a  decimal  by  10,  100,  1000,  etc., 

Remove  the  decimal  point  in  the  dividend  as  many  places  to 
the  left  as  there  are  ciphers  in  the  divisor. 

3.  If  there  is  a  remainder  after  carrying  the  work  as  far  as  de- 
sired, the  sign  ( + )  is  annexed  to  the  quotient  to  show  it  is  not  exact. 

EXAMPLES    FOR     PRACTICE. 

1.  What  is  the  quotient  of  .028  divided  by  7  ?   Ans.  .004. 

2.  What  is  the  quotient  of  .432  divided  by  .144  ?  Ans.  3, 

3.  What  is  the  quotient  of  5 15  divided  by  1.03  ?  Ans.  .500. 


142  DIVISION      OF      DECIMALS. 

4.  Divide  2.37  by  9.  Ans.  0.2633+  ;  or,  .2633^. 

5.  At  .25  dol.  a  pound,  how  much  honey  can  be  bought 
for  2.75  dollars  ? 

6.  How  many  building-lots  can  be  made  from  12.75 
acres  of  land,  allowing  .25  of  an  acre  to  a  lot? 

7.- Divide  43.12  by  10.  9.  Divide  .2806  by  1000. 

8.  Divide  7.312  by  100.      10.  Divide  734.201  by  10000. 

Perform  the  following  divisions : 

n.  57-5-4*  16.36.54-10.      ■  21.  IO-+.OI. 

12.  194-.25.         17.  3.854-100.  22.  11 4-. 11. 

13-  ■675-^- -33-       iS.  .0564-. 112.  .  23.  .114-11. 

14.   .0342  -T-. O7.        I9.    39  I. O4-MOOO.       24.   .OOO5-V-5. 

15.  .00394-. 26.    20.  246.7514-85.      25.  .00003 4- .00004. 

26.  A  farmer  sold  75  sheep  for  187.5  dollars:  what 
was  that  apiece  ? 

27.  If  you  travel  40.75  miles  in  a  day,  how  long  will 
it  take  to  travel  195.6  miles  ? 

28.  If  34.5  bushels  of  apples  cost  17.25  dollars,  what 
will  1  bushel  cost*  ? 

29.  If  18.75  tons  of  hay  cost  196.875  dollars,  what 
will  1  ton  cost  ? 

30.  How  many  revolutions  will  a  wheel  9.4  ft.  in  cir- 
cumference make  in  going  5280  feet? 

31.  If  1  acre  of  land  produces  25.6  bushels  of  corn, 
how  many  acres  will  be  required  to  produce  4635 
.bushels  ? 

32.  How  many  times  are  five  thousandths  contained 
in  37  hundredths  ? 

^.  How  many  times  are  seventy-five  ten-thousandths 
contained  in  eight  thousandths. 

34.  How  many  times  are  seven  millionths  contained 
in  three  hundred-thousandths  ? 


UNITED    STATES    MONEY. 


1 .  What  is  Money  ? 

Money  is  the  standard  of  value,  and  -is  often  called 
Currency. 

2.  What  is  United  States  Money  ? 

United  States  Honey  is  the  national  cnrrency 
of  the  United  States.    It  is  also  called  Federal  Money. 

3.  What  are  its  denominations  ? 

Eagles,  dollars,  dimes,  cents,  and  mills. 

TABLE. 

io  mills  (m.)     make  i  cent,    cL 
io  cents  "     i  dime,   d. 

io  d.,  or  ioo  cts.  "     i  dollar,  $,  or  dot 
io  dollars  "     i  eagle,  E. 

5octs.=^dol.;  33jcts.=|dol. ;  25  cts.=£dol. ; 
2octs.=|dol. ;  i2icts.=|dol.;  iocts-^-^dol. 

Notes. — 1.  It  will  be  observed  that  the  denominations  of  U.  S. 
money,  like  the  orders  of  whole  numbers,  increase  and  decrease  by 
the  scale  of  10.    It  is  thence  called  Decimal  Currency. 

2.  The  sign  of  U.  S.  money  is  the  character  ($),  called  the  dol- 
lar mark,  placed  before  the  sum  to  be  expressed. 

4 .  How  is  U.  S.  Money  written  ? 

Hollars  are  written  as  whole  numbers,  with  the 
ign  ($)  prefixed  to  them. 

Cents  are  written  in  the  first  two  places  on  the  right 
of  the  decimal  point ;  because  they  are  hundredths  of  a 
dollar.     Thus  13  dollars  25  cents  are  written  $13.25. 

Mills  are  written  in  the  third  place  on  the  right;  be- 
cause they  are  thousandths  of  a  dollar;  as  $25,038. 

Remarks. — 1.  Eacrles  are  exrresjjpd  by  tens  of  dollars  ;  dimes  by 
tens  of  cents.     TauB,  15  eagles  are  $150,  and  6  dimes  are  60  cuts. 


144         seduction    of    u.  s.  money. 

2.  As  cents  occupy  two  places,  if  the  number  to  be  expreb^ed  is 
less  than  10,  a  cipher  must  be  prefixed  to  the  figure  denoting  them. 

3.  In  business  calculations,  if  the  mills  in  the  insult  are  5  or 
more,  they  are  considered  a  cent;  if  less  than  5,  they  are  omitted. 

1.  Write  17  dollars  and  5  cents.  Ans.  $17.05. 

2.  Write  20  dollars,  10  cents,  and  3  mills.  Ans.  $20,103. 

3.  Express  3  eagles  and  4  dimes,  in  dollars  and  cents. 
Analysis. — Since  in  1  eagle  there  are  10  dollars,  in  3  E.  there 

are  3  times  10,  or  $30.     Again,  in  1  dime  there  are  10  cents,  and 
in  4  dimes  4  times  10,  or  40  cents.    Ans.  $30.40. 

4.  Write  43  dollars,  1 2  cents  and  5  mills. 
.    5.  Write  100  dollars  and  8  cents. 

6.  Write  2 1 9  dollars,  3  cents  and  4  mills. 

7.  Write  a  thousand  and  ten  'dollars  and  five  cents. 

EXERCISES    IN     READING    U.    S.   MONEY- 

5.  How  read  U.  S.  Money  ? 

Read  the  figures  on  the  left  of  the  decimal  point,  as 
dollars  ;  those  in  the  first  two  places  on  the  right,  as  cents; 
the  next  one,  as  mills  ;  the  others,  as  decimals  of  a  mill. 

Copy  and  read  the  following  sums  of  U.  S.  Money  ? 

1.  $17,213.  6.  $100.  11.  $1000.043. 

2.  $30,105.  7.  $107.  12.  $2100.05. 

3.  $42.60.  8.  $110.50.  13.  $1006.40. 

4.  $0,437.  9.  $230,061.  14.  $3050.10. 

5.  $0,805.    •  10.  $500,007.  15.  $4100.01 

REDUCTION    OF    U.   S.   MONEY. 
CASE    I. 
To  Reduce  Dollars  to  Cents  and  Mills. 
1.  How  many  cents  are  there  in  $4  ? 

Analysis. — Since  there  are  100  cents  in  a  dollar,  there  must  bo 
100  times  as  many  cents  as  dollars,  or  400  cts.    (P.  56,  Q.  17.) 


SEDUCTION     OF     U.    S.    MONET.  145 

2.  In  $6,  how  many  cents?    In  $7  ?    In  $10?    In  12? 

3.  How  many  mills  in  $6  ? 

Analysis. — There  are  iooo  mills  in  a  dollar;  hence,  in  $6 
there  must  be  iooo  times  as  many  mills  as  dols.,  or  6000  mills. 

4.  How  many  mills  in  15  cents  ?    In  52  cents  ? 

5.  In  $7,  how  many  mills  ?    In  $1 1  ?    In  $20  ? 

SLATE     EXERCISES. 

1.  How  many  cents  in  18  dollars  ? 

Analysis. — Since  there  are  100  cents  in  a  Operation. 

dollar,  there  must  be   100  times  as  many  18  dollars, 

cents  as  dollars  ;  and  100  times  18  are  1800.  IO° 

Therefore,  etc.  Ans.  ^oo  cts. 

2.  In  87  cents,  how  many  mills  ? 

Analysis.— Since   there  are  10  mills  in  87  Cents, 

a  cent,  there  must  be  10  times  as  many  mills  10 

as  cents;  and  10  times  87  are  870.  Ans  870  mills. 

6.  How  reduce  dollars  to  cents,  etc.  ? 

To  reduce  dollars  to  cents,  multiply  them  by  100. 
To  reduce  dollars  to  mills,  multiply  them  by  iooo. 
To  reduce  cents  to  mills,  multiply  them  by  10. 

Note. — To  reduce  dollars,  cents,  and  mills,  to  cents  and  mills, 
erase  the  sign  of  dollars  ($)  and  the  decimal  point 

3.  In  $40.75,  how  many  cents  ?       Ans.  4075  cents. 

4.  In  $51,073,  how  many  mills?      Ans.  51073  mills. 

Reduce  the  following  to  the  denominations  indicated? 

5.  $67  to  cents.  10.  $85.38  to  cents. 

6.  $125  to  cents.  11.  $7,375  to  mills. 

7.  $95  to  mills.  12.  $9.87!  to  mills. 

8.  $216  to  mills.  13.  8537  to  cents. 

9.  $46.10  to  cents.  14.  $1385  to  mills. 


146  REDUCTION     OF     U.    S.    MONEY. 

CASE     II. 
To  reduce  Cents  and  Mills  to  Dollars, 
i.  How  many  dollars  in  212  cents  ? 

Analysis. — Since  in  100  cents  there  is  $1,  in  212  cents  there 
are  as  many  dollars  as  100  is  contained  times  in  212  ;  and  100  is 
contained  in  212,  2  times  and  12  cents  over.     Therefore,  etc. 

2.  How  many  dollars  in  500  cents  ?    In  625  cents  ? 

3.  How  many  dollars  in  700  cents  ?    In  865  cents  ? 

4.  How  many  dollars  in  3000  mills  ? 

Solution. — As  many  as  1000  is  contained  times  in  3000,  which 
is  3  times. 

5.  How  many  dollars  in  5256  mills  ?    In  7341  mills  ? 

6.  How  many  cents  in  327  mills  ?    In  432  mills  ? 

SLATE    EXERCISES. 

1.  How  many  dollars  in  348  cents  ? 

Analysis. — Since  in  100  cents  there  is  1  dollar,  Operation. 

in  348  cents  there  are  as  many  dollars  as  there  are  I  loo)  3148 
times  100  cents  in  348  cents ;  and  ioo  is  in  348 
cents,  3  times  and  48  cents  over.     Therefore,  etc.  $3.40 

2.  How  many  dollars  in  4285  mills  ? 

Solution. — We  divide  the   given  mills  by      11000)41285 

iooo,  or  what  is  the  same  tiling,  cut  off  3  fig-  

ures  on  the  right  of  the  dividend.  Ans.  $4*285 

■7.  How  reduce  cents  and  mills  to  dollars  ? 
To  reduce  cents  to  dollars,  divide  them  by  100. 
To  reduce  mills  to  dollars,  divide  them  by  1000. 
To  reduce  mills  to  cents,  divide  them  by  10.  ' 

3.  In  235  cents,  how  many  dollars  ?        Ans.  $2.35. 

Reduce  the  following  to  the  denominations  indicated : 

4.  563  cents  to  dollars.  7.  5770  cents  to  dollars. 

5.  895  cents  to  dollars.  8.  268  mills  to  dollars. 

6.  1263  cents  to  dollars.         9.  3275  mills  to  cents. 


ADDITION     OF     U.    S.    MONET.  147 

ADDITION    OF    U.S. MONEY. 

i.  George  paid  $2.45  for  a  sled,  and  $1.63  for  a  pair 
of  skates :  what  was  the  cost  of  both  ? 

Analysis. — $2  and  $1  are  $3  ;  45  cts.  and  63  cts.  are  108  cts., 
.  or  $1.08,  which  added  to  $3,  make  $4.08.     Therefore,  etc. 

\     2.  What  is  the  sum  of  $5.17  and  $12.30  ? 

3.  If  a  hat  costs  $5.50,  and  a  vest  $9.75,  what  is  the 
cost  of  both  ? 

4.  A  farmer  sold  a  sheep  for  $5.50,  and  a.  calf  for 
$7.30 :  what  did  he  receive  for  both  ? 

5.  The  price  of  a  reader  is  87  cts.,  and  an  arithmetic 
63  cts. :  what  is  the  price  of  both  ? 

6.  If  a  man  pays  $3.25  a  day  for  board,  and  85  cents 
for  cigars,  what  are  his  daily  expenses  for  both  ? 

SLATE      EXERCISES. 
.  8.  Upon  what  principle  is  U.  S.  Money  founded  ?    '       - 
It  is  founded  upon  the  Decimal  Notation. 

9.  How  are  its  operations  performed  ? 

Its  operations  are  the  same  as  the  corresponding  opera- 
tions in  whole  numbers  and  Decimal  Fractions. 
1.  What  is  the  sum  of  $10,625;  $16,078;  $28? 

Analysis. — We  write  dollars  under  dollars,  cents  $10,625 
under  cents,  etc.,  and  beginning  at  the  right,  add  as  16.078 

in  simple  numbers,  placing  the  decimal  point  in  the  2  8.00 

amount  under  those  in  the  numbers  added.    (P.  25,  $^4.70"? 
Q.  9.) 

10.  How  add  United  States  money? 

Write  dollars  under  dollars,  cents  under  cents,  etc.,  and 
add  as  in  simple  numbers,  placing  the  decimal  point  in 
the  amount  under  those  in  the  numbers  added. 

Note. — If  any  of  the  given  numbers  have  no  cents,  their  place 
should  be  be  supplied  by  ciphers. 


148  ADDITION     OF     U.    S.    MONEY. 


(2.) 

(3.) 

(4.) 

(5-) 

$430,451 

$641,375 

$890.40 

$2056.625 

205.06 

80.06 

708.00 

140.50 

I28.OO7 

65.007 

25-56 

68.08 

ns.   763-5l8 

240.25 

7-07  . 

9-3*5 

6.  What  is  the  sum  of  $85.10 ;  $164.07 ;  and  $35.20  ? 

7.  What  is  the  sum  of  $207.56  ;  $500.65  ;  aud  $61.52  ? 

8.  Paid  $8.75  for  a  barrel  of  flour;  $5.25  for  2  barrels 
of  apples ;  and  $7  for  a  ton  of  coal :  what  was  the  amount 
of  my  bill  ? 

9.  A  farmer  bought  a  horse  for  $120,875  ;  a  yoke  of 
oxen  for  $95  ;  md  a  cart  for  $68.50 :  what  did  he  pay 
>br  all  ? 

10.  A  merchant  sold  goods  amounting  to  $150.35  to 
one  customer;  to  another  $96.40;  to  another  $110; 
and  to  another  $200.68 :  what  amount  did  he  sell 
to  all? 

11.  Add  120  dollars,  5  cents,  and  3  mills;  45  dollars, 
and  7  mills ;'  78  cents,  and  6  mills. 

12.  Add  7  dollars,  and  7  cents;  10  dollars,  and  5 
mills;  217  dollars,  and  45  cents;  and- 31  dollars. 

13.  Add  371  dollars,  40  cents,  and  8  mills;  710  dol- 
lars ;  90  dollars,  and  35  cents ;  and  219  dollars. 

14.  Add  1000  dollars;  100  dollars,  and  10  cents;  93 
cents ;  860  dollars,  and  8  cents ;  5  dollars  and  95  cents. 

15.  What  is  the  sum  of  1500  dollars  and  8  cents  + 
807  dollars,  60  cents,  and  7'  mills  +763  dollars,  3  cents 
and  5  mills  +  85  cents  and  8  mills  ? 

16.  A  lady  bought  a  dress  for  $45.63;  a  shawl  for 
$87,625  ;  a  collar  for  $15,375  ;  a  pocket-handkerchief 
for  $7.50  ;  what  was  the  amount  of  her  bill  ? 

17.  Bought  an  overcoat  for  $35.75  ;  a  dress-coat  for 
$28.62^  ;  and  a  vest  for  $9.12^ :  required  the  amount. 


SUBTRACTION     OF     U.    S.    MONEY.  149 

SUBTRACTION    OF    U.S. MONEY. 

i.  William  having  $7.62,  gave  $2.50  for  a  cap :  how 
much  did  he  have  left  ? 

Analysis. — $2  from  $7  leave  $5  ;  and  50  cents  from  62  cents 
leave  12  cents.     Therefore,  etc. 

2.  Henry  gave  a  10  dollar  bill  to  pay  for  a  hat,  the 
price  being  $7.50  :  how  much  change  should  he  receive  ? 

3.  The  price  of  a  grammar  is  85  cts..  and  that  of  a 
geography  $1.30  :  what  is  the  difference  in  their  prices? 

4.  A  father  earns  $10.75  a  week,  and  his  son  $8.50: 
how  much  more  does  the  former  earn  than  the  latter  ? 

5.  A  man  paid  $12.60  for  a  gal.  of  brandy,  and  $8.25 
for  a  bar.  of  flour :  required  the  difference  in  cost  ? 

SLATE    EXERCISES. 

I.  A  person  having  $356.07,  paid  $109,625  for  a  horse:' 
how  much  did  he  have  left  ? 

Analysis. — We  write  the  less  number  under  the  Operation. 

greater,  dollars  under  dollars,  cents  under  cents,  etc  $35^*°7 
Subtract  as  in  simple  numbers,  and  place  the  deci-        109.625 

malpoint  in  the  remainder  under  that  in  the  subtra-  $246,445 
hend.    (P.  38,  Q.  9.) 

II.  How  subtract  United  States  money? 

Write  the  less  number  under  the  greater,  dollars  under 
dollars,  cents  under  cents,  etc.,  and  subtract  as  in  simple 
numbers,  placing  the  decimal  point  in  the  remainder 
under  that  in  the  subtrahend. 

Note, — If  either  of  the  given  numbers  has  no  cents,  their  place 
should  be  supplied  by.  ciphers. 

(*•)  (3-)  (4.)  (S-) 

From   $65,875    $110.46    $68,004   $ico.oo 
Take     46.29      95-375    19.086      0.875 


150      MULTIPLICATION     OF     U.    S.    MONET. 

6.  George  gave  $1.75  for  his  geography,  and  $0,875 
for  his  arithmetic :  what  was  the  difference  in  cost  ? 

7.  A  lady  bought  articles  amounting  to  $29,375,  and 
gave  the  clerk  a  50  dollar  bill:  how  much  change  ought 
she  to  receive  ? 

8.  Bought  a  coat  for  $25.75  ;  pants  for  $14;  vest  for 
$11.50;  and  sold- wood  to  the  tailor  amounting  to  $50: 
how  much  am  I  indebted  to  him  ? 

9.  If  you  have  $407  on  deposit,  and  check  out 
$219,625,  how  much  will  you  have  left  in  bank  ? 

10.  Find  the  difference  between  $117.45  and  $201.03  ? 

11.  Find  the  difference  between  $1000  and  1000  cts.  ? 

1 2.  From  two  hundred  dollars  and  seven  cents,  take 
forty  dollars  and  5  mills. 

13.  From  one  hundred  dollars  and  6  cents,  take  five 
dollars  and  20  cents. 

14.  From  $300,  take  3  dol.,  3  cts.,  and  3  mills. 

15.  A  father  gave  one  daughter  a  music-box  worth 
$75,375,  the  other  a  sewing-machine  worth  $55.67  :  what 
was  the  difference  in  their  cost  ? 


MULTIPLICATION    OF    U.   S.  MONEY. 

1.  What  will  3  chairs  cost,  at  $7.50  each  ? 

Analysis. — 3  chairs  will  cost  3  times  as  much  as  1  chair.  Now 
3  times  $7  are  $21,  and  3  times  50  cts.  are  150  cts.,  equal  to  $1.50, 
which  added  to  $21,  make  $22.50.     Therefore,  etc.    (P.  52,  Note.) 

2.  What  cost  4  fruit-knives,  at  $2.12  apiece  ? 

3.  What  cost  5  bouquets,  at  $3.50  apiece  ? 

4.  What  cost  6  paper-folders,  at  75  cents  apiece  ? 

5.  At  $4.10  a  box,  what  will  8  boxes  of  lemons 
come  to  ? 

.  6.  At  $11.20  apiece,  what  will  10  dresses  cost? 


MULTIPLICATION     OF     U.    S.    MONEY.      151 

SLATE     EXERCISES. 

i.  What  will  1 8  ploughs  cost,  at  $13,125  apiece  ? 

Analysis.— If  1  plough  costs  $13,125,  18  ploughs  $13,125 

will  cost  18  times  as  much.     We  multiply  in  the  jg 

usual  way,  and  from  the  right  of  the  product  point 

off  three  figures  for  cents  and  mills ;  because  there  5 

are  three  places  of  cents  and  mills  in  both  factors.  _Jl__~ 

{P.  139,  Q.  28.)  $236,250 

12.  How  multiply  United  States  money  ? 

Multiply  as  in  simple  numbers,  and  on  the  right  of 
tile  product,  point  off  as  many  figures  for  cents  and  mills 
as  there  are  decimal  places  in  loth  factors. 

Note. — In  United  States  Money,  as  in  simple  numbers,  the 
multiplier  must  be  considered  an  abstract  number. 

2.  If  you  spend  87  J  cents  a  day,  what  will  you  spend 

in  7  days  ? 

Solution.— 87I  cts.=$o.875,  and  $0,875  x  7=16.125,  Arts. 


(3.) 

(4.) 

(S-) 

(6.) 

mat 

$39-35 

$60,075 

$100,008 

$82650 

By 

11 

•iS 

6.5 

•75 

7.  What  will  $6  chickens  come  to,  at  62J  cts»  each  ? 

8.  If  a  man  earns  $9.50  a  week,  what  will  be  his 
wages  for  5  2  weeks  ? 

9.  At  $1.37!  per  yard,  what  will  a  dress  containing 
20.5  yards  of  grenadine  come  to  ? 

10.  At  $7.50  a  ton,  what  will  100.5  tons  of  coal  cost  ? 

11.  What  cost  18  pianos,  at  $750  apiece  ? 

12.  What  is  the  amount  of  a  man's  expenses  for  12 
months,  if  he  spends  $86.50  a  month  ? 

13.  What  cost  25  building-lots,  at  $1250.50  a  lot? 


t 

152  DIVISION     OF     U.    8.    MONET. 

14.  At  $4.50  each  per  day,  what  will  be  the  hotel 
expenses  of  6  persons  for  4  weeks  ? 

15.  What  is  the  sum  of  12  times  87 J  cents,  and  15 
times  62-J  cents  ? 

16.  What  is  the  difference  between  20  times  $17.65, 
and  17  times  $25.40  ? 

17.  A  farmer  bought  12  calves,  at  $7.60  each ;  and  20 
sheep,  at  $4.75  each :  how  much  did  he  pay  for  both 
lots ;  and  what  is  the  difference  in  their  cost  ? 

DIVISION    OF    U.   S.   MONEY. 

1.  If  9  oranges  cost  6$  cents,  what  will  1  orange  cost  ? 
Analysis.— If  9  oranges  cost  63  cents,  1  orange  will  cost  1 

ninth  of  63  cts.,  which  is  7  cts.     Therefore,  etc.    (P.  63,  Q.  10.) 

2.  If  7  sheep  cost  $35,  what  will  one  cost  ? 

3.  If  8  yards  of  velvet  cost  $72,  what  will  1  cost  ? 

4.  A  man  laid  out  $50  in  vests,  which  were  $5  apiece: 
how  many  did  he  buy  ? 

5.  How  many  hand-carts,  at  $6,  can  be  bought  for  $300  ? 

SLATE    EXERCISES. 

1.  Sold  6  hats  for  $42.75 :  what  was  that  apiece  ? 

ANALYSIS. — 1  hat  is  1  sixth  of  6  hats ;  hence  1        Operation. 
hat  is  worth  1  sixth  of  $42.75,  which  is  $7,125.  6)$42-75° 

The  object  of  this  example  is  to  divide  the  sum     Ans.  $7.1 25 
O'  $42.75   into  6  equal  parts.      (P.  63,  Q.    10.) 
Dividing  as  in  simple  numbers,  there  is  a  Temainder  of  3  cents 
.  which  we  reduce  to  mills,  and  dividing  as  before,  point  off  threa 
figures  for  cents  and  mills. 

13.  How  divide  United  States  money  by  an  abstract  number  ? 

Reduce  the  dividend  to  mills,  and  divide  as  in  simple 
numbers.  The  quotient  will  he  mills,  which  must  be  re- 
duced to  dollars  and  cents. 

Note. — If  there  is  a  romaindor,  write  tho  sign  +  after  tho 
quotient. 


DIVISION    OF    U.    S.    MONEY.  153 

2.  How  many  hats,  at  $7,125,  can  be  bought  for 

$42.75  ? 

Analysis.— The  object  here  is  to  find  Operation. 

how  many  times  one  sum  of  money  is  con-     $7. 1 2 5) $42.7 5 0(6 
tained  in  another.    But  the  divisor  contains  42.750 

dollars,  cents,  anr?  mills,  while  the  dividend 
contains  dollars  and  cenU  only.     We  there- 
fore reduce  the  latter  to  mills,  and  then  divide  as  in  simple  num- 
bers.   (P.  67,  Q.  10.) 

13,  a.  How  divide  U.  S.  Money  by  U.  S.  Money  ? 

Reduce  the  divisor  and  dividend  to  the  same  denomina- 
tion, and  divide  as  in  simple  numbers.  Tlie  quotient 
will  be  times,  or  an  abstract  number.     (P.  63,  Q.  10.) 

Notes.— 1.  In  business  matters  it  is  rarely  necessary  to  carry 
the  quotient  beyond  mills. 

2.  If  there  is  a  remainder  after  all  the  figures  are  divided, 
annex  ciphers,  and  continue  the  division  as.  far  as  desirable,  con- 
sidering the  ciphers  annexed  as  decimals  of  the  dividend. 

(30  (4.)  (50  (6.) 

7)$q2.6q4  8)$ii4  9)  $13,791         6)$o.8o4 

$13,242  $14.25  $1,421+  $0,134 

7.  How  many  times  are  $8  contained  in  $90.47  ? 

8.  How  many  times  are  $75  contained  in  $900  ? 

9.  How  many  melons,  at  $0.25  each,  can  be  purchased 

/3r  $15.75  ? 

10.  How  many  ponnds  of  butter,  at  $0.30,  can  be  had 
for  $25.65? 

11.  If  I  pay  $14,875  for  7  baskets  of  peaches,  how 
much  is  that  a  basket  ? 

12.  A  stationer  sold  5  slates  for  $0,625:  what  was 
that  apiece  ? 


154  QUESTIONS      FOR     REVIEW. 

13.  At  $20  apiece,  how  many  yearlings  can  be  bought 
for  $280  ? 

14.  How  many  acres  of  land,  at  $12.50  per  acre,  can 
you  buy  for  $1000  ? 

15.  Paid  $43  for  8  excursion-tickets:  how  much  was 
that  for  each  ticket  ? 

16.  A  clerk  agreed  to  work  12  months  for  $427.56: 
what  was  that  per  month  ? 

17.  If  $1600.75  are  divided  equally  among  25  per- 
sons, how  much  will  each  receive  ? 

18.  Sold  280  sheep  for  $658 :  what  was  that  per  head  ? 

19.  Sold  35  doz.  eggs  for  $8.75  :  what  was  that  a  doz.  ? 

20.  Paid  $2675.75  for  278  tons  of  coal:  what  was  the 
cost  per  ton  ? 

QUESTIONS     FOR     REVIEW. 

1.  A  man  bought  6  cords  of  wood  at  $4.17,  and  5  tons 
of  coal  at  $7,375  :  what  did  he  pay  for  both  ? 

2.  If  you  buy  10  pen-knives  for  $6.75,  and  sell  them 
at  85  cents  apiece,  how  much  will  you  make  or  lose  ? 

3.  What  is  the  difference  between  7  times  $8.50,  and 
9  times  $7,625  ? 

4.  What  is  the  difference  between  11  times  $17.65, 
and  8  times  $19.48? 

5.  Bought  8  boxes  of  raisins  at  $6.40,  and  sold  them 
at  $9.63  a  box :  how  much  was  made  on  them  ? 

6.  A  traveler  was  robbed  of  $375,  and  had  $159.60 
left:  how  much  money  had  he  before  the  robbery? 

7.  If  you  buy  12  melons  for  $3.96,  and  sell  them  at  50 
cents  each,  how  much  will  you  make  on  each  ? 

8.  A  grocer  bought  12  bags  of  coffee  for  $121.92,  but 
finding  it  damaged  sold  it  for  $30.20  less  than  cost:  for 
what  did  he  sell  it  per  bag  ? 


APPLICATIONS     OF     IT.    S.    MONEY.        155 

9.  Bought  in  barrels  of  flour  for  $897.50:  for  how 
much  must  I  sell  it  per  barrel  to  make  $300  ? 

io.  Paid  $1162.50  for  25  acres  of  land:  what  is  that 
^>er  acre  ? 

n.  Paid  $6785  for  100  oxen:  what  was  that  apiece  ? 

12.  How  many  horses,  at  $150,  will  $10650  buy  ? 

13.  What  is  the  sum  and  difference  of  $567,625  and 

*945-5°  t 

14.  How  many  tubs  of  butter,  at  $16.50  each,  can  be 
bought  for  $206.25  ? 

15.  Exchanged  75  barrels  of  apples  worth  $150,  for 
25  barrels  of  cider :  what  did  the  cider  cost  per  barrel  ? 

16.  How  many  pair  of  shoes,  at  $1  J,  can  be  had  for 
5000  pounds  of  rice,  at  12^  cents  a  pound  ? 


APPLICATIONS    OF    U.   S.   MONEY. 
BILLS. 

14.  What  is  a  bill? 

A  Sill  is  a  written  statement  of  goods  sold,  services 
rendered,  etc.,  and  should  include  the  various  items,  the 
price  of  each,  the  date,  and  the  place  of  the  transaction. 

15.  How  are  bills  receipted  ? 

A  Hill  is  receipted  when  it  contains  the  words, 
"Received  payment,"  and  is  signed  by  the  person  to 
whom  it  is  due,  or  his  agent. 

16.  What  is  the  meaning  of  the  terms  Debtor  and  Creditor  ? 
A  debtor  is  a  person  who  owes  a  debt. 

A  creditor  is  one  to  whom  a  debt  is  owed. 

Notes. — 1.  The  abbreviation  Dr.  denotes  debit  or  debtor  ;  Cb., 
credit  or  creditor  ;  'per  signifies  by,  and  the  character  @,  at. 

2.  To  familiarize  the  learner  with  the  form  of  bills,  the  manner 
of  receipting  them,  etc.,  it  is  advisable  for  him  to  copy  the  follow- 
ing, in  a  neat  hand,  upon  his  slate  or  paper. 


156        APPLICATIONS     OF     U.    S.    MONEY. 

Find  the  amount  of  the' following  bills : 

(i.)  New  YonK,June  3d,  1871. 

Hon.  GrEORGE  Peabody, 

Bought  of  Horace  Webster. 
7  lbs.  coffee,  @  $0.38      ....    $2.66 

1.92 

.66 

4-35 


12    "    sugar,  "       .16 

6    "    cornstarch,    "       .11 
5    «    tea,  "       .87      ..    .      . 

Amour* 
Received  Payment, . 

Hoeace  Webster. 

(2.)  Mobile,  Feb.  21st,  1871. 

George  Walker,  Esq., 

To  Daniel  Kingsbury  &  Co,  Dr. 
To  15  yds.  silk,  @  $2.35 
"11  *  muslin,  "  .29 
"  T2  pair  hose,  "  .42 
"12  "  gloves,  "  1.50 
"     6  parasols,  "       3.50 

Amt, 
Redd  Paft  by  Note, 

D.  Kingsbury  &  Co., 

By  S.  Barret, 

(3.)  Chicago,  May  21st,  1871. 

John  Murdoch,  Esq.,  in  acct.  tvith  David  Joy  &  Co. 

Dr. 

For  12  pair  shoes,  @    $1.62 

"      6    "    thick  boots,       *       2.75 

"    10     "    slippers,  "         .88 

"    24    "    woollen  hose,    "         .30 

Carried  forward,  $51.94 


APPLICATIONS     OF     U.    S.    MONEY.        157 


Brought  forward, 

§51.94 

Credit, 

By    6  bushels  wheat, 

@ 

$1.50 

"14       "       oats, 

a 

.60 

"      3  barrels  cider, 

a 

2.75 

"    10  barrels  potatoes, 

a 

I.87 

What  is  the  balance  ? 

Balance^ 

44.35 

,  $7.59 

(4.) 

Messrs.  J.  H.  Burtis  &  Co., 

Bought  Of  SCHERMERHORN  &   WlLSON. 

12  slates  @  $.13  ;  3  blackboards  @  $9.50;  6  boxes  of 
crayons  @  $.68 ;  36  inkstands  @  $.12  ;  2  small  globes 
@  $5.25.     Eequired  the  amount. 

(5.) 

James  Barber  in  acct,  with  W.  C.  Young. 

Dr. 

For  10  shovels  %  $1.67 ;  12  hoes  @  1.25  ;  6  rakes  @ 

$1.50;  4  axes  ®  $2.63.  ; 

Credit. 

By  12  days  work,  man,  @  $2.00;  10  days  work,  self 
and  team,  @  $3.60;  20  cords  wood  @  $3.10;  15  shade- 
trees  @  $1.75. 

What  is  the  balance  due  on  the  above  ? 

6.  George  Morris  of  Chicago  bought  of  A.  T.  Stewart 
&  Co.  18  yds.  of  silk,  at  $2.63 ;  14  yds.  Empress  cloth, 
at  $1.75  ;  12  yds.  of  poplin,  at  $2,375  ;  15  yds.  of  French 
lawn,  at  $0.65;  6  pair  of  gloves,  at  $1.85;  6  pocket- 
handkerchiefs,  at  $1.25;  3  parasols,  at  $3.60.  Eequired 
the  amount. 


COMPOUND     NUMBERS. 


1.  What  are  Simple  Numbers? 

Simple  Numbers  are  those  which  contain  units 
of  one  denomination  only; "  as,  two,  four,  3  apples, 
4  quarts,-  etc. 

2.  What  are  Compound  Numbers  ? 

Compound  Numbers  are  those  which  contain 
two  or  more  denominations  of  the  same  natter e ;  as, 
4  bushels  and  3  pecks ;  3  days  and  5  hours. 

Illustration. — Suppose,  for  example,  we  apply  the  inch  as  a 
unit  of  measure  to  the  side  of  a  table,  and  find  it  equal  to  30  such 
measures.  Again,  if  we  employ  the  foot  (12  in.)  as  the  unit,  it  is 
equal  to  two  such  measures,  and  6  in.  oyer.  Now  as  6  inches  — 
\  foot,  we  may  call  its  length  2\  feet,  or  2  feet,  6  inches.  The 
former  expression  contains  units  of  but  one  denomination,  viz., 
feet ;  therefore,  it  is  a  simple  number.  The  latter  contains  units 
of  two  different  denominations,  which  are  of  the  same  nature,  viz., 
feet  and  inches  ;  therefore  it  is  a  compound  number. 

But  the  expression  2  feet  and  4  pounds  is  not  a  compound 
number;  for  the  units  are  of  unlike  natures. 

Note.  —  Compound  numbers  are  often  called  Denominate 
Numbers. 

3.  When  it  is  said  that  a  cane  is  3  feet  long,  what  is  the  kind 
of  number  used  ? 

A  Simple  Number ;  because  it  contains  but  one 
denomination,  viz.,  feet. 

4.  If  we  say  that  a  cane  is  2  feet  and  10  inches  long,  what  kind 
is  the  number  ? 

A  Compound  Number  ;  because  it  contains  tivo 
denominations  of  the  same  nature,  viz.,  feet  and  inches. 

What  kind  of  a  number  is  6  days  ?     Why  ? 

What  kind  is  5  pounds  2  shillings  and  6  pence?  Why? 

What  kind  is  each  of  the  following:  Two?  Three? 
12  oranges?  7  pounds  and  5  ounces?  10  dollars  and 
25 .cents?     10  oxen? 


COMPOUND     NUMBERS.  159 

UNITED    STATES    MONEY. 
5.  What  is  United  States  Money  ? 

United  States  Money  is  the  national  currency 
of  the  United  States,  and  is  often  called  Federal  Money. 
J.  What  are  its  denominations  ? 

Eagles,  dollars,  dimes,  cents,  and  mills* 

TABLE. 
10  mills  (m.)      are     i  cent,  ct 

10  cents  "       i  dime,  d. 

10  d.,  or  ioo  cts.  "       i  dollar,  $,  or  dot 
10  dollars  "       i  eagle,  E. 

7.  Of  how  many  kinds  is  U.  S.  money  ? 
Two,  Paper  and  Metallic. 

8.  What  is  the  'paper  money  of  the  U.  S.  ? 

The  Paper  Money  of  the  U.S.  consists  of  Treasury- 
notes issued  by  the  Government,  known  as  Greenbacks, 
and  Bank-notes  issued  by  Banks. 

Note. — Paper  money  is  called  Paper  Currency. 

Treasury-notes  less  than  $i  are  called  Fractional  Currency. 

9.  What  is  metallic  money  ? 

3fetallic  Money  consists  of  stamped  pieces  of 
metal,  called  coins.  It  is  also  called  specie,  or  specie 
currency. 

Note. — For  exercises  in  U.  S.  money  see  pp.  143-157. 

10.  Of  what  do  the  coins  of  the  United' States  consist  ? 
Gold  coins,  silver  coins,  and  the  minor  coins. 

11.  Name  the  coins  of  each. 

The  gold  coins  are  the  double-eagle,  eagle,  half -eagle, 
quarter-eagle,  three-dollar,  and  dollar  piece. 

The  silver  coins  are  the  trade  dollar,  half-dollar, 
quarter-dollar,  twenty-cent  piece,  and  dime. 

The  minor  coins  arc  the  nickel  $-cent  and  ycent 
pieces,  and  the  bronze  -cent. 


160  COMPOUND     NUMBERS. 

ENGLISH     MONEY. 

12.  What  is  English  money? 

English  Money  is  the  national  currency  of  Great 
Britain,  and  is  often  called  Sterling  Money. 

13.  What  are  the  denominations? 
Pounds,  shillings,  pence,  and  farthings. 

TABLE. 

4  farthings  (qr.  or  far.)  are  i  penny,  d. 

\2  pence  "    i  shilling,  s. 

20  shillings  "    i  pound  or  sovereign,  £. 

21  shillings  "    i  guinea,  •  g. 

Notes. — i.  The  legal  value  of  a  pound  Sterling,  or  sovereign, 
Is  $4.8665;  the  value  of  an  English  shilling  is  24^  cents;  and  that 
of  a  penny  about  2  cents. 

2.  Farthings  are  commonly  expressed  as  fractions  of  a  penny. 
Thus,  1  far.=^d.;  2  far.=£d. ;  3  far.=$d. 

1.  How  many  farthings  in  5  pence  ? 

Analysis. — Since  there  are  4  farthings  in  a  penny,  there  must 
be  4  times  as  many  farthings  as  pence,  and  4  times  5  are  20  pence. 
Therefore,  etc. 

2.  How  many  pence  in  3  shillings  ?    In  5  s.  ?    In  7  s.  ? 

3.  How  many  shillings  in  £3?    In  £5  ?    In  £10  ? 

4.  In  15  farthings  how  many  pence  ? 

Analysis. — Since  in  4  farthings  there  is  1  penny,  in  15  far. 
there  are  as  many  pence  as  4  far.  are  contained  times  in  15  far  ;l 
and  4  is  in  15,  3  times  and  3  over.  Therefore,  in  15  far.  there  are 
3d.  and  3  far.  over,  or  3^d. 

5.  How  many  shillings  in  18  pence  ?  .  In  24d.  ?  . 

6.  How  many  pounds  in  21  shillings?  In  30  s. ? 
In  40  s.  ?    In  85  s.?    In  100  s.  ? 

\*  If  the  teacher  desires  further  practice  upon  the  Tables,  as 
they  are  recited,  he  will  find  corresponding  Examples  in  the  Slate 
Exercises,  pp.  173-178. 


COMPOUND     NUMBERS.  161 

TROY    WEIGHT. 

1 4.  For  what  is  Troy  Weight  used  ? 
For  weighing  gold,  silver,  and  jewels. 

15.  What  are  the  denominations  ? 

Pounds,  ounces,  pennyweights,  and  grains. 

TABLE. 

24  grains  (gr.)      are  1  pennyweight,    pwt. 
20  pennyweights    a    1  ounce,  oz. 

12  ounces  "    1  pound,  lb. 

Note. — The  best  method  of  imparting  to  children  a  correct 
idea  of  Weights  and  Measures,  is  to  let  them  see  and  handle  the 
actual  standards,  or  some  material  objects  which  are  equal  to  the 
several  units  of  length,  surface,  capacity,  and  weight.  In  this  way, 
the  Compound  Tables  afford  a  wide  field  for  object  teaching. 

1.  How  many  grains  in  2  pennyweights? 

2.  How  many  pennyweights  in  3  ounces  ?    In  5  oz.  ? 

3.  How  many  ounces  in  3  pounds  ?     In  5  pounds  ? 

4.  How  many  ounces  in  40  pwt.  ?    In  45  pwt.  ? 


AVOIRDUPOIS     WEIGHT. 

16.  For  what  is  Avoirdupois  weight  used? 

For  weighing  all  coarse  articles;    as,  hay,  cotton 
groceries,  etc.,  and  all  metals  except  gold  and  silver. 

17.  What  are  the  denominations  ? 
Tons,  hundreds,  pounds,  and  ounces. 

TABLE. 

16  ounces  {oz)  are  1  pound,  lb. 

100  pounds  "    1  hundred  weight,  ciot. 

20  cwt.,  or  2000  lbs.,   "    1  ton,  T. 

8  oz.  =  J  lb. ;  4  oz.  =  J  pound. 


162  COMPOUND     NUMBEES. 

Notes. — i.  The  ounce  is  often  divided  into  halves,  quarters,  etc. 

2.  In  business  transactions,  the  dram,  the  quarter  of  25  lbs., 
and  the  firkin  of  56  lbs.,  are  not  used  as  units  of  Avoirdupois 
Weight. 

3.  Net  weight  is  the  weight  of  goods,  without  the  bag,  cask,  etc. 

4.  Gross  weight  is  the  weight  of  goods  with  the  bag,  cask,  or 
box  in  which  they  are  contained.  It  calls  28  lbs.  a  quarter,  112 
pounds  a  hundred  weight,  and  2240  pounds  a  long  ton. 

1.  How  many  ounces  in  3  pounds ?    In  4  lbs.? 

2.  How  many  pounds  in  3  quarters  ?     In  5  qrs.  ? 

3.  How  many  hundreds  in  3  tons  ?     In  5  tons  ? 

4.  In  40  ounces,  how  many  pounds  ?    In  48  oz.  ? 

5.  In  30  hundreds,  how  many  tons  ?    In  60  cwt.  ? 

6.  In  3500  pounds,  how  many  tons?    In  5000  lbs. ? 


APOTHECARIES'    WEIGHT. 

1 8.  For  what  is  Apothecaries'  Weight  used  ? 
For  mixing  medicines. 

19.  What  are  the  denominations  ? 

Pounds,  ounces,  drams,  scruples,  and  grains. 

TABLE. 

20  grains  (gr.)  are  1  scruple,  sc,  or  3. 

3  scruples  "    1  dram,      dr.,  or  3 . 

8  drams  "    1  ounce,     oz.,  or  f . 

12  ounces  "    1  pound,                g>. 

Note. — The  only  difference  between  Troy  and  Apotliecaries3 
weight,  is  in  the  division  of  the  ounce.  The  pound,  ounce,  and 
grain  are  the  same  in  each. 

1.  How  many  grains  in  2  scruples  ?    In  3  scruples  ? 

2.  How  many  scruples  in  4  drams  ?    In  8  drams  ? 

3.  How  many  drams  in  5  ounces  ?    In  100  ounces  ? 

4.  How  many  ounces  in  6  pounds  ?    In  1 2  pounds  J 


COMPOUND     NUMBERS.  163 

LINEAR    MEASURE. 

20.  For  what  is  Linear  Measure  used  ? 

For  measuring  that  which  has  length,  without  breadth; 
as,  lines,  distances,  etc.    It  is  often  called  Long  Measure. 

21.  What  are  the  denominations  ? 

Leagues,  miles,  furlongs,  rods,  yards,  feet,  and  inches, 

TABLE. 

ix  inches  {in.)        are  i  foot,  •       ft. 

3  feet  "    i  yard,  yd. 

5  \  )rds.,  or  1 6  \  ft.      "    i  rod,  perch,  or  pole,  r.,  or  p. 
40  rods,  "    1  furlong,  fur. 

8  fur.,  or  3 20  rods  "    i  mile  m. 

3  miles  "    1  league,  I. 

Note. — The  inch  is  commonly  divided  into  halves,  fourths, 
eighths,  or  tenths  ;  sometimes  into  twelfths,  called  lines. 

1.  Draw  a  straight  line  2  inches  long  on  your  slate  or 
blackboard. 

2.  Draw  one  4  in.  long.  6  in.  long.  9  in.  long.  A  foot 
long.    A  yard  long. 

3.  How  long  is  your  pencil  ?  This  pen-knife  ?  This 
pen-holder?    This  paper-folder ?     This  ruler? 

4.  How  long  is  this  table  ?  How  wide  ?  How  long 
is  the  school-room  ?  How  wide  ?  How  high  ?  How 
long  is  the  play-ground  ?  * 

5.  How  many  inches  in  3  feet  ?    In  5  feet  ?    In  8  feet  ? 

6.  How  many  inches  in  2  ft.  and  5  in.  ?    4  ft.  and  6  in.  ? 

7.  How  many  feet  in  4  yards  ?    In  7  yds.  ?    In  9  yds.  ? 

8.  How  many  feet  in  5  yards  and  2  feet  ?  In  6  yds. 
and  4  ft.  ? 

*  This  exercise  should  be  varied,  and  continued  till  the  class 
obtain  a  clear  idea  of  the  ordinary  measures  of  length. 


164  COMPOUND     NUMBBBB. 

9.  How  many  miles  in  5  leagues  ?    In  8  leagues  ? 

10.  How  many  feet  in  2  rods?    In  3  rods? 
•    In  37  inches,  how  many  feet?    In  60  in.?    In 

?     In  100  in.? 

In  1 8  feet,  h  ow  many  yards  ?  In  2  8  ft.  ?    In  40  ft.  ? 

In  32  furlongs,  how  many  miles?    In  41  fur.? 

fur.? 


11. 

75  in.  ?     In  100  in.: 
12 

In  50  fur, 


CLOTH     MEASURE. 

22.  For  what  is  Cloth  Measure  used? 

For  measuring  those  articles  of  commerce  whose 
length  only  is  considered;  as,  cloths,  laces,  ribbons,  etc. 

23.  What  are  the  denominations  ? 

The  Linear  Yard  is  the  principal  unit.  This  is 
divided  into  quarters,  eighths,  and  sixteenths. 

TABLE. 

3  ft.  or  36  in->  are  1  yard,    -    -    -       yd. 
18  in.,  "    1  half  yard,  -    -    i  yd. 

9  in.,  "    1  quarter  yard,  -    J  yd. 

4-J  in.,  .    "    1  eighth       "    -    I  yd. 

2\  in.,  "    1  sixteenth,  "    -  TV  yd. 

Note. — Ells  Flemish,  English,  and  French,  are  no  longer  used 
iu  the  United  States ;  and  the  nail,  as  a  measure,  is  practically 
obsolete. 

1.  How  many  quarters  in  14  in.  ?    In  26  in.  ? 
2i  How  many  fourths  in  3J  yards  ?    In  5 J  yards  ? 

3.  How  many  eighths  in  25  in.  ?     In  37  in.  ? 

4.  How  many  eighths  in  2|  yards  ?    In  3I  yards  ? 

5.  How  many  yards  in  14  half  yards  ?  In  30  halves  ? 
In  35  halves  ? 

6.  How  many  yards  in  25  fourths  of  a  yard  ?  In  32 
fourths  ?    In  48  sixteenths  ? 


COMPOUND     NUMBERS, 


165 


SQUARE     MEASURE. 

24.  For  what  is  Square  Measure  used  ? 

For  measuring  surfaces,  or  that  which  has  length  and 
breadth,  without  thickness;  as,  land,  flooring,  etc.  It  is 
often  called  Land  Measure. 

25.  What  are  the  denominations? 

Acres,  square  rods,  square  yards,  square  feet,  and 
square  inches. 


TABLE. 

144  square  in.  (sq.  in.) 

are    1  square  foot, 

sq.ft. 

9  square  feet 

"      1  square  yard, 

sq.  yd. 

30 J  square  yards,  or  ) 
2 72 J  square  feet,         ) 

u  j  1  sq.  rod,  perch, 
1     or  pole, 

sq.  r. 

160  sq.  rods 

"      1  acre, 

A. 

640  acres 

"     1  square  mile, 

M. 

Note. — The  acre  was  formerly  divided  into  4  roods ;  but  in 
practice  the  rood  is  no  longer  used  as  a  unit  of  measure. 

26.  What  is  a  Square  ? 

A  Square  is  a  rectilinear  figure 
which  has/owr  equal  sides  and  four 
right  angles.    Thus, 

A  Square  Inch  is  a  square, 
each  side  of  which  is  1  inch  in 
length. 

A  Square  Yard  is  a  square, 
each  side  of  which  is  1  yard  in 
length. 

Note. — The  corners  of  any  square  figure,  also  of  a  table,  a 
room,  etc.,  are  right  angles. 

1.  Make  a  right  angle  upon  your  slate,  or  the  black- 
board. 


3eq.  ft.x3=9pq.  ft. 

9  eq.  ft.  =  i  eq.  yd. 


166  COMPOUND     NUMBERS. 

2.  Make  a  square  inch. 

3.  Make  a  square  whose  side  is  3  inches.    6  inches. 

4.  Make  a  square  foot. 

5.  Make  a  square  yard. 

6.  Divide  a  square  yard  into  square  feet. 

7.  Divide  a  square  foot  into  square  inches. 

8.  How  many  square  inches  in  2  sq.  ft.  ?     In  3  sq.  ft.  ? 

9.  How  many  square  feet  in  3  sq.  yds.  ?    In  5  sq.  yds.  ? 

10.  In  27  sq.  feet,  how  ma*hy  sq.  yards  ?   In  $6  sq.  feet  ? 

11.  What  is  the  difference  between  3  feet  square,  and 
3  square  feet  ? 


CUBIC     MEASURE. 

27.  For  what  is  Cubic  Measure  used  ? 
For  measuring  solids  ;  as,  timber,  boxes  of  goods,  the 
capacity  of  rooms,  ships,  etc.    It  is  often  called  Solid 


2§.  What  are  the  denominations  ? 

Cords,  cubic  yards,  cubic  feet,  and  cubic  inches. 

TABLE. 

1738  cubic  inches  (cu.  in.)  are  1  cubic  foot,  cu.ft. 

27  cubic  feet  "   1  cubic  yard,  cu.  yd. 

128  cubic  feet  "   1  cord,  C.        1 

28,  a.  Describe  a  cord  ?    A  foot  of  wood  ? 

A  Cord  of  wood  is  a  pile  8  ft.  long,  4  ft.  wide,  and  4 
ft.  high  :  for,  8  x  4  x  4=  128. 

A  Cord  Foot  is  one  foot  in  length  of  such  a  pile ; 
hence,  8  cord  feet  make  one  cord. 

Note. — Timber  is  now  measured  by  cubic  feet  and  inches. 

The  old  cubic  ton  of  40  feet  of  round  timber,  and  50  feet  of 
hewn  timber,  has  fallen  into  disuse  in  the  United  States. 


COMPOUND     NUMBERS. 


16? 


27  cu.  ft.  =  1  cu.  yd. 


29.  What  is  a  Cube? 

A  Cube  is  a  regular  solid 
bounded  by  six  equal  squares, 
called  its  faces.    Thus, 

A  Cubic  Inch  is  a  cube, 
each  side  of  which  is  a  square 
inch. 

A  Cubic  Yard  is  a  cube, 
each  side  of  which  is  a  square 
yard. 

1.  Draw  a  cubic  inch. 

2.  Draw  a  cube  whose  sides  are  3  inches  square. 

3.  Draw  a  cubic  foot. 

4.  How  long  and  wide  must  a  block  of  marble 
whose  height  is  3  feet,  to  form  a  cubic  yard  ? 

5.  How  many  cubic  feet  in  2  cubic  yards  ? 

6.  How  many  cubic  yards  m  54  cubic  feet  ? 

7.  In  2  cords,  how  many  cubic  feet  ? 


be, 


DRY     MEASURE. 

30.  For  what  is  Dry  Measure  used  ? 

For  measuring  grain,  fruit,  salt,  etc. 

31.  What  are  the  denominations  ? 

Chaldrons,  bushels,  pecks,  quarts,  and  pints. 

TABLE. 

2  pints  (pt.)  are  1  quart,        qt. 

8  quarts  "   1  peck,         ph. 

4  pecks,  or  32  qts.,  "   1  bushel,       bu. 

36  bushels  "   1  chaldron,  ch. 

Notes. — 1.  The  dry  quart  is  equal  to  i\  liquid  quart  nearly. 
2.  The  chaldron  is  used  for  measuring  coke  and  bituminous  coal. 


1 


168  COMPOUND     NUMBERS. 

i.  In  8  pints,  how  many  quarts  ?     In  16  pints  ? 

2.  In  32  quarts,  how  many  pecks  ?     In  40  quarts  ? 

3.  How  many  pecks  in  5  bushels  ?    In  7  bushels  ? 

4.  How  many  quarts  in  5  pecks  ?    In  9  pecks 

5.  How  many  bushels  in  12  pecks?    In   17  pecks 

6.  How  many  quarts  in  2  bushels  and  3  pecks  ? 

7.  How  many  quarts  in  3  pecks  and  4  quarts  ? 

8.  How  many  bushels  in  40  quarts  ?    In  64  quarts  ? 

LIQUID     MEASURE. 

32.  For  what  is  Liquid  Measure  used  ? 

For  measuring  milk,  wine,  vinegar,  molasses,  etc. 

33.  What  are  the  denominations  ? 

Hogsheads,  barrels,  gallons,  quarts,  pints,  and  gills. 

TABLE. 

4    gills  (gi.)  are  1  pint,  pt. 

2    pints  "    1  quart,  qt. 

4    quarts  "    1  gallon,  gal. 

31^  gallons  "    1  barrel,  bar.,  or  bbl. 

63    gallons  ■    1  hogshead,         hlid. 

Notes. — 1.  Liquid  Measure  is  often  called  Wine  Measure. 
2.  The  old  Beer  Measure  is  practically  obsolete  in  this  country 

1.  How  many  gills  in  4  pints  ?    In  10  pints  ? 

2.  How  many  pints  in  7  quarts  ?    In  9  quarts  ? 

3.  How  many  quarts  in  5  gallons?    In  10  gallons? 

4.  In  20  quarts,  how  many  gallons  ? 

5.  In  24  pints,  how  many  quarts? 

6.  In  16  gills,  how  many  pints? 

7.  In  24  gills,  how  many  pints  ?    How  many  quarts  ? 

8.  In  32  pints,  hoW  many  quarts  ?   How  many  gallons.? 

9.  How  many  gallons  in  2  barrels  ? 
10.  How  many  gallons  in  2  hogsheads  ? 


COMPOUND     NUMBEKS. 


169 


CIRCULAR     MEASURE. 

34.  For  what  is  Circular  Measure  used  ? 

For  measuring  angles,  land,  latitude  and  longitude, 
the  motion  of  the  heavenly  bodies-,  etc. 

35.  What  are  the  denominations  ? 
Signs,  degrees,  minutes,  and  seconds. 


TABLE 


6o  seconds  (")      are  i  minute, 

6o*minutes  "    i 

30  degrees 

12  signs,. or  3600 

Note. — Signs  as  a  measure  are  used  only  in  Astronomy 


1  sign,  s. 

1  circumference,   cir. 


36.  What  is  a  Circle  ? 
A    Circle   is  a   plane   figure 

bounded  by  a  curve  line,  every 
part  of  which  is  equally  distant 
from  a  point  within  called  the 
center. 

37.  What  is  the  Circumference  of 
a  Circle  ? 

The  Circumference  of  a  Circle  is  the  curve 
line  by  which  it  is  bounded.     It  is  divided  into  3600. 

38.  What  is  the  Diameter? 

The  Diameter  is  a  straight  line  drawn  through  the 
centre,  terminating  at  each  end  in  the  circumference. 

39.  What  is  the  Radius  ? 

The  Radius  is  a  straight  line  drawn  from  the 
center  to  the  circumference,  and  is  equal  to  half  the 
diameter. 

8 


170  COMPOUND     NUMBERS. 

40.  What  is  an  Arc  of  a  Circle  ? 

An  Arc  of  a  Circle  is  any  part  of  the  circum- 
ference. 

In  the  preceding  figure,  ADEBFis  the  circum- 
ference; A  B  the  diameter;  C  A,  C  D,  C  E,  etc.,  are 
radii ;  A  D,  D  E,  etc.,  are  arcs. 

Draw  a  circle.     Draw  a  diameter.     Draw  another  1 
diameter  perpendicular  to  the  first. 

Note. — These  two  diameters  divide  the  circumference  into  four 
equal  parts,  called  quadrants. 

41.  How  many  degrees  in  a  quadrant  ? 

Ninety. 

42.  How  many  and  what  angles  do  these  two  diameters  form  ? 

Four  right  angles. 

43.  How  many  degrees  in  a  right  angle  ? 

Ninety. 

MEASUREMENT    OF    TIME. 

44.  What  are  the  denominations  of  Time  ? 

Centuries,  years,  months,  tveeks,  days,  hours,  minutes, 
and  seconds. 

TABLE. 

6o  seconds  (sec.) 
6o  minutes 
24  hours 
7  days 

365  days, 
52  w. 

366  days 
12  calendar  months  (mo.) 

100  years 

Note. — In  most  business  transactions,  30  days  are  considered 
a  month.    Four  weeks  are  sometimes  called  a  lunar  month. 


ays,  or        ) 
r.  and  1  d.,  ) 


are 

i  minute, 

mm. 

a 

1  hour, 

h. 

« 

1  day, 

d. 

u 

1  week, 

w. 

a 

1  common  year, 

c.y. 

tt 

1  leap  year, 

E* 

t( 

1  civil  year, 

y- 

a 

1  century, 

ce*„ 

COMPOUND     NUMBERS. 


171 


45.  What  is  a  common  year  ? 

A  common  year  is  one  which  contains  365  days. 

46.  What  is  a  solar  year  ? 

A  solar  year  is  the  time  in  which  the  earth  revolves 
around  the  sun,  and  equals  365  d.  5I1. 48  min.  and  49.7  sea 

Note.— The  excess  of  the  solar  above  the  common  year  is 
about  6  hours,  or  \  of  a  day,  nearly ;  hence,  in  4  years,  it  amounts 
to  about  1  day. 

47.  What  is  a  leap  year? 

A  leap  year  is  one  which  contains  366  days. 

48.  How  caused,  and  why  so  called  ? 

It  is  caused  by  the  excess  of  a  solar  above  a  common 
year ;  and  is  so  called  because  it  leaps  over  the  limit,  or 
runs  on  one  day  more  than  a  common  year. 

This  day  is  given  to  February,  because  it  is  the  short- 
est month;  hence,  in  leap  years,  February  has  29  days. 

49.  What  is  a  civil  year  ? 

A  civil  year  is  the  year  adopted  by  government 
for  computing  time,  and  includes  both  common  and 
leap  years  as  they  occur. 

50.  How  is  the  civil  year  divided  ? 

It  is  divided  into  1 2  calendar  months,  viz. : 
January       {Jan.),    the  first  month,  has  3 1  days. 


February 

(mi 

"   second 

a 

28 

a 

March 

{Mar.), 

"  third 

a 

31 

a 

April 

{Apr.), 

"  fourth 

a 

30 

a 

May 

{May), 

"  fifth 

it 

31 

it 

June 

{June), 

"  sixth 

it 

30 

a 

July 

{July), 

"  seventh 

a 

31 

a 

August 

{Aug.), 

"  eighth 

a 

31 

a 

September 

{Sept.), 

"  ninth 

a 

30 

a 

October 

{Oct.), 

"  tenth 

it 

31 

a 

November 

{Nov), 

"  eleventh 

a 

30 

a 

December 

{Dec), 

«  twelfth 

u 

31 

t( 

172 


COMPOUND     NUMBERS, 


Note. — The  following  couplet  will  aid  the  leaner  In  remem- 
bering the  months  that  have  30  days  each : 
"  Thirty  days  hath  September, 
April,  June,  and  November." 

Each  of  the  other  months  has  31  days,  except  February,  which 
in  common  years  has  28  days,  but  in  leap  years,  29. 

51.  Into  how  many  seasons  is  the  year  divided? 

Four,  viz.:  Spring,  Summer,  Autumn,  and  Winter. 

52.  Name  the  months  of  each  season  ? 
Spring  consists  of  March,  April,  and  May. 
Summer         "        June,  July,  and  August. 
Autumn        *        September,  October,  and  November. 
Winter  "        December,  January,  and  February. 


MISCELLANEOUS 

12  things  are  1  dozen. 
12  doz.       "    1  gross. 

24  sheets  are  1  quire  of  paper. 
2c  quires   "  1  ream    " 

2  leaves     are 


TABLES. 

1 2  gross  are  1  great  gross. 
20  things   "  1  score. 

2  reams    are  1  bundle. 
5  bundles    "  1  bale. 

1  folio. 

1  quarto,  or  4to. 
1  octavo,  or  8vo. 
1  duodecimo,  or  12  mo. 
1  eighteen  mo. 
1  twenty-four  mo. 
Note. — The  terms  folio,  quarto,  octavo,  etc.,  denote  the  nurrfim- 
of  leaves  into  which  a  sheet  of  paper  is  folded  in  making  books. 

Aliquot  Parts  of  a  Dollar,  or  100  Cents. 


4  leaves 

8  leaves 

12  leaves 

18  leaves 

24  leaves 


50  cents  as  $}. 
33I  cents  ss  %\. 
25  cents  ss  $J. 
20  cents  =  %\. 
i6f  cents  =  %\. 


12J  cents  ==  $|. 

10  cents  as  %^. 
8-J  cents  =  %JV 
6J  cents  =  $tV 
5    cents  =  $-2*ff. 


REDUCTION 


1.  What  is  Reduction  ? 

Reduction  is  changing  a  number  from  one  denom- 
ination to  another,  without  altering  its  value.  It  is  of 
two  kinds,  Ascending  and  Descending. 

2.  What  is  Reduction  Descending? 

Reduction  Descending  is  changing  higher  de- 
nominations to  lower  j  as,  feet  to  inches,  etc. 

3.  What  is  Reduction  Ascending  ? 

Reduction  Ascending  is  changing  lower  de- 
nominations to  higher;  as,  inches  to  feet,  etc. 

To  Reduce  Higher  Denominations  to  Lower. 

i.  How  many  farthings  are  there  in  £16,  5  s.  4d.  2  far.  ? 

Analysis. — Since  there  are  20s.  in  a  Operation. 

pound,  there  must  be  20  times  as  many        £16,  5s.  4d.  2  far. 
shillings  as  pounds,  plus  the  given  shil-  2o 

lings.     Now  20  times  16  are  320s.,  and         

3203. +  5s.=325s.    Again,  since  there  are         **3 

I2d.  in  a  shilling,  there  must  be  12  times       

as  many  pence  as  shillings,  plus  the  given       3 9046L 

pence.    Now  12  times  325  are  3900,  and  4 

390od.  +  4d.=3go4d.    Finally,  since  there    — ~z  „         . 

are  4  far.  in  a  penny,  there  are  4  times  as       5         Iar«  A-ns. 

many  farthings  as  pence,  plus  the  given 

farthings.     Now  4  times  3904  are  15616  far.,  and  15616  far.  + 

2  far.— 15618  far.     Therefore,  etc. 

4.  How  reduce  higher  denominations  to  lower  ? 

Multiply  the  highest  denomination  hy  the  number  re- 
quired  of  the  next  lower  to  make  one  of  the  higher,  and 
to  the  product  add  the  lower  denomination. 

Proceed  in  this  manner  with  the  successive  denomina- 
tions, till  the  one  required  is  reached. 


.174  REDUCTION". 

2.  Reduce  £5,  6s.  9^d.  to  farthings  ? 
Suggestion.— £5,  6s.  g£d.  =  £5,  6s.  gd.  2  far.    Arts.  5126  far. 

3.  Eeduce  £9,  is.  6Jd.  to  farthings  ? 

4.  In  17s.  4d.  3  far.,  how  many  farthings  ? 

5.  In  £43,  4s.  how  many  pence  ? 

6.  In  £115,  how  many  farthings  ? 

To  Reduce  Lower  Denominations  to  Higher. 

7.  Eeduce  156 18  farthings  to  pounds  ?. 

Analysis. — Since  in  4  farthings  there  is  Operation. 

1  penny,  in  15618  far.  there  are  as  many      a\  15618  far. 

pence  as  4  is  contained  times  in  15618  ;  and         

15618  divided  by  4=3904^  and  2  far.  over.      I2  )  39°4d.  2  far„ 
Again,  since  in  I2d.  there  is  is.,  in  3go4d.        2Q\       -g     ^ 

there  are  as  many  shillings  as  12  is  con- —1 

tained  times  in  3904;  and  3904-7-12=3255.  £16,  5s. 

and  4d.  over.    Finally,  in  325s.  there  &re  Am.  £16,  5S.4d.  2f. 

as  many  pounds  as  20  is  contained  times  in 

325  ;  and  325-f-20=£i6  and  5s.  over.     Therefore,  etc. 

5.  How  reduce  lower  denominations  to  higher  ? 

Divide  the  given  denomination  by  the  number  required 
of  this  denomination  to  make  a  unit  of  the  next  higher. 

Proceed  in  this  manner  with  the  successive  denomina- 
tions, till  the  one  required  is  reached.  The  last  quotient, 
with  the  several  remainders  annexed,  ivill  be  the  answer. 

Note. — The  remainders,  it  should  be  observed,  are  the  same 
denomination  as  the  respective  dividends  from  which  they  arise 
(P.  62,  Q.  5,  Rem.) 

Pkoof. — Reduction  Ascending  and  Descending  prove 
each  other ;  for  one  is  the  reverse  of  the  other. 

8.  Reduce  1231  pence  to  pounds?    Ans.  £5,  2s.  7d. 
Proof. — Reversing  the  operation  we  have  20         £5,  2S.  7<L 

times  5  =  ioos.,  and   ioos.  +  2s.=iG2s.     Again,  20 

12  times  io2  =  i224d.,  and  I224d.  +  7d.=i23id.,        j~g, 

the  same  as  the  given  number  of  pence.    There-  j  2 

fore,  the  work  is  right.  , 

1231& 


REDUCTION.  175 

9.  In  1 46 1  pence,  how  many  pounds,  shillings,  and 
pence  ? 

10.  In  27035  farthings,  how  many  pounds,  etc.  ? 

11.  What  will  25  pen-knives  cost,  at  2s.  6d.  apiece? 

Solution — 2s.  6d.=3od.  Now  30CL  x  25  =  7500!.,  and  75od.= 
£3,  23.  6d. 

1 2.  What  will  75  slates  cost,  at  1 1  pence  each  ? 

13.  How  many  pennyweights  in  7  lb.  5  oz.  troy  ? 

14.  How  many  grains  in  10  lb.  7  oz.  6  pwt.  9  gr.  ? 

15.  Eeduce  156 1  pwt.  to  pounds  and  ounces  ? 

16.  Eeduce  6575  grains  to  pounds,  etc. 

17.  A  goldsmith  made  12  gold  rings,  each  weigh- 
ing 3  pwt.  4  gr. :  how  many  ounces  of  gold  did 
he  use? 

18.  A  lady  bought  a  gold  chain  weighing  2  oz.  12  pwt., 
at  $1.50  a  pennyweight:  how  much  did  she  pay  for  her 
chain  ? 

19.  Eeduce  2265  ounces  Avoirdupois  to  hundreds. 

20.  In  15  T.  2  cwt.  31  lb.  8  oz.,  how  many  ounces  ? 

21.  Eeduce  100  tons,  75  lb.  4  oz.  to  ounces. 

22.  What  will  5  lbs.  4  oz.  of  candy  come  to,  at  6  cts. 
an  ounce  ? 

23.  A  farmer  sold  2  tons,  375  lbs.  of  maple  sugar,  at 
15  cts.  a  pound :  how  much  did  he  receive  for  it  ? 

24.  In  5  lb.,  apothecaries'  weight,  how  many  drams  ? 

25.  In  7  lb.  4  oz.,  how  many  scruples  ? 

26.  In  469  scruples,  how  many  apothecaries'  pounds? 

27.  In  1578  grains,  how  many  apothecaries'  ounces? 

28.  How  many  feet  in  45  rods  ? 

Note. — For  multiplying  by  5^  or  i6£,  the  number  of  yards  or 
feet  in  a  rod,  see  Note,  p.  113. 

29.  How  many  feet  in  12  miles,  10  rods,  and  7  feet  ? 

30.  How  many  yards  in  26  miles,  3  fur.  2  yards  ? 


170  REDUCTION. 

31.  In  456  yards,  how  many  rods  ? 

Remake. — To  divide  by  5  A-  or  i6£  (the  Operation. 

number  of   feet  or  yards  in  a  rod)  we  5^)  45  6  yd. 

reduce  both  the  divisor  and  dividend  to  2           2 

halves ;    then  divide   in    the  usual  way.  —     

Thus,  5i=n  halves,  and  456=912  halves;  n    )  912 

now  11  is  contained  in  912,  82  times  and  *    "    7T              j 

10  remainder.    But  the  dividend  is  half 

yards ;  therefore  the  10  remainder  is  half  yards,  and  is  equal  to 

5  yards.    (P.  120,  Q.  45.) 

32.  In  1560  feet,  how  many  rods  ? 

^2,.  How  many  miles,  rods,  etc.,  in  11278  feet? 

34.  In  5  1.  17  m.  3  fur.  6  r.  10  ft.,  how  many  feet  ? 

35.  When  the  fare  is  5  cts.  a  mile,  what  will  it  cost 
you  to  ride  1 5  leagues  ? 

36.  How  many  rods  of  fence  are  required  on  both 
sides  of  a  road  2  miles  long  ? 

37.  Allowing  a  military  step  to  be  2-J  ft,  how  many 
steps  will  a  soldier  take  in  marching  5  miles  ? 

38.  How  many  quarters  of  a  yard  in  45  inches  ? 

39.  How  many  eighths  of  a  yard  in  27  inches  ? 

40.  Keduee  151  yards  to  sixteenths. 

41.  In  951  eighths  of  a  yard,  how  many  yards? 

42.  If  you  pay  8  cts.  for  \  yard  of  muslin,  how  much 
would  you  have  to  pay  for  20  yards  ? 

43.  A  lady  paid  $3  for  J  yard  of  lace;  what  would  a 
piece  of  35  yards  come  to  at  that  rate  ? 

44.  How  many  square  feet  in  160  sq.  rods  ? 

45.  How  many  sq.  feet  in  5  A.  61  sq.  rods? 

46.  How  many  sq.  yd.  in  21  A.  36  sq.  rods? 

47.  How  many  sq.  rods  in  a  sq.  mile  ? 

48.  In  85 1  sq.  rods,  how  many  acres  ? 

49.  In  75625  sq.  yards,  how  many  acres? 

50.  In  46273  sq.  inches,  how  many  sq.  yards? 


REDUCTION.  177 

51.  What  will  be  the  cost  of  a  village-lot  containing 
20  sq.  rods  of  land,  at  25  cts.  per  sq.  foot  ? 

5  2  What  will  it  cost  to  sod  a  park  of  2  acres,  at  1 2 
cts.  a  sq.  yard  ? 

53.  How  many  cu.  in.  in  41  cu.  yards,  16  cu.  feet? 

54.  How  many  cu.  yards  in  96365  cu.  inches  ? 

55.  Eeduce  4250  cu.  feet  to  cords  ? 

56.  Eeduce  75^  cords  to  cu.  feet  ? 

57-  Eeduce  15  cords  and  2^  cu.  feet  to  cu.  feet? 

58.  At  4  cts.  a  cu.  foot,  what  will  it  cost  to  excavate 
a  cellar  containing  450  cu.  yards  ? 

59.  What  will  12  cords  of  wood  come  to,  at  5  cts.  a 
cubic  foot  ? 

60.  What  is  the  worth  of  168  cord  feet  of  wood,  at 
$4  a  cord  ? 

61.  Eeduce  6  bu.  3  pk.  5  qt.  to  pints  ? 

62.  How  many  bushels  in  1647  quarts  ? 

6$.  What  cost  5  oushels  of  chestnuts,  at  12  cts.  a 
quart  ? 

64.  A  lad  bought  2  bushels  of  apples  for  $2.50,  and 
sold  them  at  40  cts.  a  half  peck :  what  was  his  profit  ? 

65.  How  many  quart  boxes  are  required  to  hold  4  bu. 
1  pk.  of  blackberries  ? 

66.  At  6  cts.  a  quart,  how  many  bushels  of  peanuts' 
can  be  bought  for  $6.42  ? 

67.  How  many  gills  in  6  gal.  1  qt.  1  pt.  3  gills? 

68.  How  many  quarts  in  3  hhd.  3  gal.  2  qt.  ? 

69.  In  1832  gills,  how  many  gallons  ? 

70.  In  2560  quarts,  how  many  hogsheads? 

71.  A  milkman  having  a  15  gallon  can  full  of  milk, 
sold  15  quarts,  and  spilt  the  rest :  how  many  quarts  did 
he  lose  ? 

72.  What  cost  a  hogshead  of  maple  syrup,  at  25  cents 
a  quart  ? 


178  REDUCTION. 

73.  A  druggist  paid  $126  for  a  cask  of  alcohol  con- 
taining 42  gal.,  and  sold  it  at  20  cts.  a  gill :  how  much 
did  he  make  ? 

74.  How  many  seconds  in  3  days,  5  hr.  17  minutes? 

75.  How  many  days  in  4565  minutes  ? 

76.  How  many  minutes  in  7  weeks,  5  days  ? 

77.  How  many  minutes  in  a  common  year? 

78.  How  many  common  years  in  48256  hours? 

79.  If  a  clock  ticks  seconds,  how  many  times  does  it 
tick  in  a  week  ? 

80.  At  $3.50  per  day,  what  will  it  cost  me  to  board 
12  weeks? 

81.  If  a  man's  pulse  beats  73  times  a  minute,  how 
many  times  will  it  beat  in  3 1  days  ? 

82.  If  a  steamer  sails  11  miles  an  hour,  how  long  will 
it  take  her  to  sail  3000  miles  ? 

83.  Eeduce  45 °  13'  to  seconds. 

84.  How  many  degrees  in  10000"? 

85.  How  many  signs  in  8275'? 

86.  The  earth  revolves  3600  on  its  axis  in  24  hours : 
how  many  degrees  does  it  revolve  in  1  hour  ?  How 
far  in  4  minutes  ? 

87.  How  many  sheets  of  paper  in  5  reams,  10  quires? 

88.  How  many  reams  in  12258  sheets  ? 

89.  If  you  pay  $2.50  a  ream  for  paper,  what  is  that  a 
sheet  ? 

90.  How  many  crayons  are  there  in  40  boxes,  each 
containing  a  gross  ? 

91.  What  will  25  gross  of  lead  pencils  cost,  at  42  cts. 
a  dozen  ? 

92.  Pens  are  packed  in  boxes  containing  a  gross:  how 
many  pens  are  there  in  6  boxes  ? 

93.  A  man  having  100  dozen  eggs,  sent  them  to  mar- 
ket in  16  boxes :  how  many  eggs  did  he  put  in  a  box  ? 


RECTANGULAR     SURFACES.  179 

MEASUREMENT  OF  RECTANGULAR 
SURFACES. 

6.  What  is  a  Rectangular  Figure  ? 

A  Rectangular  Figure  is  one  which  has  four 
sides  and  four  right  angles.     (See  next  Fig.) 

When  all  the  sides  are  equal,  it  is  called  a  square , 
when  the  opposite  sides  only  are  equal,  it  is  called  an 
oblong,  or  parallelogram. 

7.  What  is  the  area  of  a  figure  ? 

The  Area  of  a  Figure  is  the  quantity  of  surface 
it  contains.     It  is  often  called  the  superficial  contents. 

Note. — The  term  rectangular  signifies  right  angled. 

i.  How  many  square  feet  of  canvas  in  a  rectangular 
painting  3  feet  long  and  2  feet  wide  ? 

Illustration. — Let  the  painting  be  rep- 
resented by  the  figure  in  the  margin  ;  its 
length  being  divided  into  three  equal  parts, 
and  its  breadth  into  two ;  each  denoting  a 
linear  foot.  It  will  be  seen  that  there  are 
2  rows  of  squares  in  the  figure,   and   3  3  feet' 

squares  in  a  row.    Therefore,  the  painting  must  contain  2  times  3, 
or  6  square  feet. 

8.  How  find  the  area  of  a  rectangular  surface  ? 
Multiply  the  length  and  breadth  together. 

2.  How  many  square  feet  in  a  blackboard  8  ft.  long 
and  i>i  ft.  wide  ? 

3.  How  many  square  inches  in  a  pane  of  glass  32  in. 
long  and  24  in.  wide  ? 

4.  How  many  square  rods  in  a  garden  15  rods  long 
and  6  rods  wide  ? 

5.  How  many  yards  of  carpeting  1  yard  wide  are 
required  to  cover  a  room  18  feet  long  and  15  feet  wide  ? 


180 


RECTANGULAR     SOLIDS. 


6.  How  many  sq.  feet  in  a  board  16  ft.  long  and  1}  ft 
wide  ? 

7.  How  many  acres  in  a  farm  100  rods  long  and  80 
rods  wide  ? 

8.  How  many  acres  in  a  township  6  miles  square  ? 

9.  A  flower  garden  is  30  yards  long  and  18  yards  wide- 
what  are  its  contents  ? 

10.  How  many  brick  8  in.  long  and  4  in.  wide,  will  it 
iake  to  pave  a  sidewalk  40  ft.  long  and  5  ft.  wide  ? 

11.  What  is  the  cost  of  a  pine  board  18  ft.  long  and 
2 \  ft.  wide,  at  8  cts.  a  square  foot  ? 


MEASUREMENT     OF     RECTANGULAR 
SOLIDS. 


4  feet. 


~^ 


P 


9.  What  is  a  Rectangular  Body  ? 

A  Rectangular  Body 

is  one  bounded  by  six  rectan- 
gular sides,  each  opposite  pair 
being  equal  and  parallel;  as, 
boxes  of  goods,  blocks  of  hewn 
stone,  etc. 
When  all  the  sides  are  equal,  it  is  called  a  cube. 

10.  What  are  the  Contents  of  a  body? 

The  Contents  or  Solidity  of  a  body  is  the  quai* 
tity  of  matter  or  space  it  contains. 

1.  How  many  cu.  feet  are  there  in  a  box  of  books  4  ft. 
long,  3  ft.  wide,  and  2  ft.  deep  ? 

Illustration. — Let  the  box  be  represented  by  the  preceding 
figure  ;  its  length  being  divided  into  four  equal  parts,  its  breadth 
into  three,  and  its  depth  into  tiw  ;  each  part  denoting  a  linear 
foot.     In  the  upper  surface  of  the  box  there  are  3  time*  4,  or  12 


RECTANGULAR     SOLIDS.  181 

pq.  feet.  Now,  if  the  box  were  i  foot  deep,  it  would  contain  i 
time  as  many  cubic  feet  as  there  are  square  feet  in  its  upper  face, 
and  i  time  4  x  3=12  cu.  ft.  But  the  box  is  2  fe^t  deep  ;  therefore 
it  must  contain  2  times  4  x  3=24  cu.  feet. 

11.  How  find  the  contents  of  a  rectangular  body. 
Multiply  the  length,  breadth,  and  thickness  together. 

2.  How  many  cu.  inches  in  a  Lrick  8  in.  long,  4  in. 
wide,  and  2  in.  thick  ? 

3.  How  many  cubic  feet  in  a  box  of  sugar  5  ft.  long, 

3  ft.  wide,  and  3  ft.  deep  ? 

4.  Henry  made  a  pile  of  cubic  letter  blocks,  the 
length  of  which  was  8  blocks,  the  width  6  blocks,  and 
the  height  5  blocks :  how  many  blocks  were  in  the  pile  ? 

5.  How  many  cubic  feet  in  a  pile  of  brick  13  ft.  long, 
7  ft.  wide,  and  5  ft.  high  ? 

6.  How  many  cubic  feet  in  a  load  of  wood  7  ft.  long, 

4  ft.  wide,  and  3 \  ffc.  high  ? 

7.  How  many  cu.  feet  in  a  bin  12  ffc.  long,  6  ft.  wide, 
and  5  \  ft.  deep  ? 

8.  How  many  cu.  yards  of  earth  must  be  removed 
to  dig  a  cellar  36  ft.  long,  20  ft.  wide,  and  6  ft.  deep  ? 

9.  How  many  cu.  feet  in  a  stick  of  timber  36  ft.  long, 
1 J  ft.  wide,  and  i|  ft.  thick  ? 

10.  What  will  it  cost  to  build  a  wall  120  ft.  long,  1}  ft. 
thick,  and  9  ft.  high,  at  15  cts.  a  cubic  foot? 

11.  What  will  it  cost  to  dig  a  trench  100  ft.  long,  9  ft 
wide,  and  4^  ft.  deep,  at  27  cts.  a  cu.  yard  ? 

12.  What  is  the  worth  of  a  pile  of  wood  48  ft.  long, 
6  ft.  high,  and  4  ft.  wide,  at  $4}  a  cord  ? 

13.  A  rectangular  bin  is  10  ft.  long,  6  ft.  wide,  and 
4  feet  deep :  what  are  its  contents  ? 

14.  A  load  of  wood  is  7-|  feet  long,  4  ft.  wide,  and  3  ft. 
high :  what  are  its  contents  ? 


182  DENOMINATE    FEACTIONS. 

REDUCTION    OF    DENOMINATE    FRACTION& 

12.  What  is  a  Denominate  Fraction? 

A  Denominate  Fraction  is  one  or  more  of  the 
equal  parts  into  which  a  Compound  or  Denominate 
number  may  be  divided. 

13.  How  are  they  expressed  ? 

Denominate  Fractions  are  expressed  either  as 
common  fractions,  or  as  decimals;  as,  \  pound,  .8  yard. 

To  reduce  Denominate  Fractions  to  Units  of  Lower 
Denominations. 

i.  Reduce  f  gallon  to  quarts  and  pints. 

Analysis. — Since  there  Operation. 

are  4  qts.  in  a  gal.,  there  J  g.  x  4=f,  or  1  qt.  and  f  qt.  rem. 
must  be  4  times  as  many  3  qt.  X  2  =  6    or  I1  t)t. 

quarts  as  gallons;  and  4  ■    Am,  \  1   xi   *t    ' 

times  £  gal.  are  § ,  equal  x         ■  r 

to  1  qt.  and  £  qt.  rem.     Again,  since  there  are  2  pt.  in  a  quart, 
there  must  be  2  times  f  or  f  pt.,  equal  to  i£  pt.     Therefore,  eic. 

11.  How  reduce  denominate  fractions  to  units  of  a  lower 
denomination  ? 

1.  Multiply  the  given  numerator  by  the  number  required 
to  reduce  the  fraction  to  the  next  lower  denomination, 
and  divide  the  product  by  the  denominator.  (P.  173,  Q.  4.) 

II.  Multiply  and  divide  the  successive  remainders  in  the 
same  manner  till  the  lowest  denomination  is  reached.  The 
several  quotients  will  be  the  answer  required. 

2.  Reduce  |  of  a  yard  to  feet  and  inches. 

3.  Reduce  f  of  a  pound  sterling  to  shillings  and  pence. 

4.  Reduce  fa  of  a  week  to  days  and  hours. 

5.  What  part  of  a  pint  is  fa  of  a  gallon  ? 

Solution. — This  example  is  the  same  in  principle  as  the  pre- 
ceding. Reducing  the  numerator  to  the  required  denomination, 
place  it  over  the  givon  denominator  :  /*  gal.  x  4  x  2— J$,  or  |  pt. 


DENOMINATE     FRACTIONS.  183 

6.  What  part  of  a  quart  is  ^  of  a  bushel  ? 

7.  What  part  of  a  pennyweight  is  ^-J^  pound  Troy  ? 

8.  Eeduce  .6  yard  to  feet  and  inches. 

Analysis. — Reducing  .6  yard    to   feet,  we  have         -6  jd. 

.6  yd.  x  3  =  1   ft.   and  .8   ft.   over.     Again,   reducing  _^ 

.8  ft.  to  inches,  we  have  .8  ft.  x  12=9.6  in.     There-  1.8  ft. 
fore,  .6  yard  equals  1  ft.  9.6  in.,  which  is  the  answer        I2 

required.  Ans.  1  ft.  9.6  in.  9.6  in. 

15.  How  reduce  a  denominate  decimal  to  units  of  lower 
denominations  ? 

I.  Multiply  the  denominate  decimal  by  the  number  re- 
quired of  the  next  lower  denomination  to  make  one  of 
the  given  denomination,  and  point  off  the  product  as  in 
multiplication  of  decimals. 

II.  Proceed  in  this  manner  with  the  decimal  part  of  the 
successive  products,  as  far  as  required.  The  integral 
part  of  the  several  products  will  he  the  answer. 

Note. — The  preceding  operations  in  Denominate  Fractions  are 
the  same  in  principle  as  those  in  Reduction  Descending. 

9.  Reduce  .84  gal.  to  quarts  and  pints. 

10.  Reduce  .625  week  to  days,  etc. 

11.  Reduce  .875  bushel  to  pecks,  etc. 

To  reduce  a  Compound  Number  from  a  Lower  to  a  Denonv 
inate  Fraction  of  a  Higher  Denomination. 

12.  What  part  of  a  gallon  is  1  pint  and  2  gills  ? 
Analysis. — 1  pt.  2  gi.  —  6  gills ;  1  pt.  2  gi.  =  6  gi. 

and  1  gallon=i  x  4  x  2  x  4=32  gills.      r  gal.  x  4  X  2  X  4  =  32  gi. 
But  6  gills  are  &  of  32  gi,  equal  to  ,  6  3       <« 

-h  gal.  Therefore,  etc.  MS'  *  °r  T  *  gaL 

16.  How  reduce  a  compound  number  to  a  denominate  fraction 
of  a  higher  denomination  ? 

I.  Reduce  the  given  number  to  its  lowest  denomination 
for  the  numerator. 

II.  Reduce  to  the  same  denomination,  a  unit  of  the 
required  fraction,  for  the  denominator. 


i.5  pt. 


184  DENOMINATE     FRACTIONS. 

Note. — If  the  lowest  denomination  of  the  given  number  con- 
tains a  fraction,  the  number  must  be  reduced  to  the  parts  indi- 
cated by  the  denominator  of  the  fraction.    (Ex.  16.) 

13.  Reduce  2  ft.  5  in.  to  the  fraction  of  a  yard. 

14.  Reduce  3  qt.  1  pt.  to  the  fraction  of  a  bushel. 

15.  What  part  of  a  pound  sterling  is  12s.  6d.  ? 

1 6.  What  part  of  a  pound  Troy  is  f  pennyweight  ? 
Solution — The  lowest  denomination  is  sths  of  a  pwt.    Now 

1  lb.  Troy = 1  x  12  x  20  x  5  =  1200  fifths  pwt.    Ana.  t^ott*  or  Tou  lb. 

17.  What  part  of  a  mile  is  j  of  a  rod  ? 

18.  Reduce  f  of  a  quart  to  the  fraction  of  a  bushel  ? 

19.  What  decimal  part  of  a  gallon  is  3  qt.  1  pt.  2  gi.? 
Analysis. — Since  4  gi.  are  1  pt.,  there  Operation. 

must  be  1  fourth  as  many  pints  as  gills,  and  4    2  gi. 

i  of  2  gi.=f,  or  .5  pc.     Place  the  .5  pt.  on 

the  right  of  the  given  pints.     Again,  2  pt. 

are  1  qt. ;  hence,  there  is  1  half  as  many 

quarts  as  pints;  and  £  of  1.5  pt.=.75  qt., 

which  we  place  on  the  right  of  the  given     Ans.  O.g-IJK  °"al. 

quarts.     Finally,  4  qt.  are  1  gal. ;   hence, 

there  is  1  fourth  as  many  gallons  as  quarts;  and  J-  of  3.75  qt. 

=0.9375  gal.     Therefore,  etc. 

17.  How  reduce  a  compound  number  to  a  denominate  decimal 
of  a  higher  denomination. 

I.  Write  the  numbers  in  a  column,  placing  the  lowest 
denomination  at  the  top. 

II.  Beginning  with  the  lowest,  divide  it  by  the  number 
required  of  this  denomination  to  make  a  unit  of  the  nex-t 
higher,  and  annex  the  quotient  to  the  next  higher. 

Proceed  in  this  manner  with  the  successive  denomina- 
tions, till  the  one  required  is  reached. 

20.  Reduce  3  fur.  20  rods  to  the  decimal  of  a  mile. 

21.  What  decimal  of  a  pound  is  6s.  8d.  ? 

22.  Reduce  5  gal.  3  qt  to  the  decimal  of  a  hogshead. 


3-75  qt. 


COMPOUND    ADDITION 


Operation. 

Gal. 

qt. 

pt. 

gi. 

5 

3 

I 

3 

8 

2 

I 

I 

4 

3 

I 

3 

i.  What  is  the  sum  of  5  gal.  3  qt.  1  pt.  3  gi.;  8  gal. 
2  qt,  1  pt.  1  gi.;  4  gal.  3  qt.  1  pt.  3  gi.? 

Analysis. — Write  the  numbers  so 
that  the  same  denominations  shall 
stand  in  the  same  column,  and  begin- 
ning at  the  right,  add  the  columns  sep- 
arately. Thus,  3  gi.  and  1  gi.  are  4  gills, 
and  3  are  7  gills,  equal  to  1  pt.  and  3  gi. 
Set  the  3  gi.  under  the  column  of  gills,  Ans.  19  2  o  3 
and  carrying  the  1  pt.  to  the  column  of 

pints,  the  sum  is  4  pints,  equal  to  2  qts.  and  no  remainder.  Write 
a  cipher  under  the  pints,  and  carrying  the  2  qts.  to  the  column 
of  quarts,  the  sum  is  10  qts.,  equal  to  2  gal.  and  2  quarts.  Set  the 
2  qts.  under  the  quarts,  and  carrying  the  2  gal.  to  the  column  of 
gallons,  the  sum  is  19  gals.  Hence,  19  gal.  2  qt.  o  pt.  3  gi.  is  the 
sum  required. 

1.  How  add  Compound  Numbers  ? 

I.  Write  the  same  denominations  one  under  another, 
and  beginning  at  the  right,  add  each  column  separately. 

II.  If  the  sum  of  a  column  is  less  than  a  unit  of  the 
next  higher  denomination,  write  it  under  the  column  added. 

If  equal  to  one  or  more  units  of  the  next  higher 
denomination,  carry  these  units  to  that  denomination, 
wd  write  the  excess  under  the  column  as  in  Simple  Ad- 
dition.    (P.  28,  Q.  13.) 

Remake. — Addition,  Subtraction,  etc.,  of  Compound  Numbers 
are  the  same  in  principle  as  the  corresponding  operations  in 
Simple  Numbers.  The  only  difference  between  them  arises  from 
their  scales  of  increase.  The  orders  of  the  latter  increase  by  the 
constant  scale  of  10.  The  denominations  of  the  former  increase  by 
a  variable   scale.     In  both  we  carry  for  the  number  which  it 


186  COMPOUND     ADDITION. 

takes  of  a  lower  order  or  denomination  to  make  one  in  the  next 
higher.  In  the  former,  this  number  is  always  10 ;  in  the  latter, 
it  is  variable. 

(4  (3.)  (4.) 

£.        s.       d.    far.  Lb.     oz.    pwt.  Yd.    ft.     in. 

10       13      4      2  13       8      9  825 

7        5      3      3               8      6      8  7      1      & 

8352                583  527 


(5-) 

(6.) 

(7.) 

Bu.  pk.  qt. 

pt. 

T. 

cwt. 

lb. 

M. 

fur.  r. 

15   3   7 

1 

5 

18 

35 

35 

3  28 

30  2   3 

1 

8 

3 

83 

84 

5   is 

8  3   5 

0 

3 

17 

35 

38 

3   8 

53  1   6 

1 

7 

5 

70 

27 

4  13 

8.  In  one  bin  there  are  35  bu.  3  pk.  and  7  qts.  of 
oats;  in  another,  27  bu.  2  pk.  5  qts.;  and  in  another, 
28  bu.  3  pk:  how  many  bushels  are  there  in  all? 

9.  A  merchant  sold  3  pieces  of  muslin  :  one  contain- 
ing 35  yds.  and  1  fourth ;  another,  43  yds.  and  5  eighths ; 
and  the  other,  38  yds.  and  3  eighths:  how  many  yards 
did  he  sell  ? 

10.  In  one  garden  there  are  13  sq.  r.  5  sq. yd.  3  sq.ft.; 
in  another,  18  sq.  r.  8  sq.  yd.  5  sq.  ft.;  and  in  another, 
23  sq.  r.  5  sq.  yd.  and  8  sq.  ft:  how  much  land  in  all? 

1 1.  A  farmer  has  4  fields :  one  containing  18  A.  35  sq. 
rods;  another  30  A.  78  sq.  r. ;  another  45  A.  30  sq.  r., 
the  other,  23  A.  65  sq.  r. :  how  many  acres  has  he? 

12.  How  much  wood  is  there  in  3  loads,  one  of  which 
contains  1  0.  45  cu.  ft.;  another,  1  C.  and  58  cu.  ft; 
and  the  other  1  0.  85  cu.  ft.  ? 


COMPOUND    SUBTRACTION. 


i.  From  27  yd.  i  ft.  8  in.,"  take  18  yd.  2  ft.  5  in. 

Analysis. — Write  the  less  number  under  Operation. 

the  greater,  placing  the  same  denominations  Yd.     ft.     in. 

in  the  same  column.     Beginning  at  the  right,  27      I      8 

we  proceed  thus :  5  in.  from  8  in.  leave  3  in. ;  182      S 

set  the  3  under  the  column  of  inches.     Next, 

since  2  ft.  cannot  be  taken  from  1  ft.,  we  bor-  Ans.  823 
row  a  unit  of  the  next  higher  denomination,  . 
which  is  yards.  Now  1  yd.  or  3  ft.  added  to  1  ft.  make  4  ft.,  and 
2  ft.  from  4  ft.  leave  2  ft.  Finally,  1  to  carry  to  18  makes  19,  and 
19  yds.  from  27  yds.  leave  8  yds.  Hence,  the  difference  is  8  yds. 
2  ft.  3  in. 

2.  How  subtract  Compound  Numbers  ? 

I.  Write  the  several  denominations  of  the  subtrahend 
under  those  of  the  same  name  in  the  minuend. 

II.  Beginning  at  the  right,  subtract  each  denomination 
of  the  subtrahend  from  that  above  it,  and  set  the  remain- 
der under  the  term  subtracted. 

III.  If  the  number  in  any  denomination  of  the  subtra- 
hend is  larger  than  that  above  it,  add  to  the  upper  num- 
ber as  many  as  are  required  to  make  a  unit  of  the  next 
higher  ;  then  subtract  and  carry  1  to  the  next  denomina- 
tion in  the  subtrahend,  as  in  Simple  Subtraction. 

(*•)  (3-) 

From  £25,  7s.  6d.  2  far.         13  lb.  7  oz.  18  pwt.  23  gr. 
Take  $23,  5s.  3d.  3  far.  7  lb.  8  oz.  13  pwt.  18  gr. 

4.  From  2  bu.  take  3  pk.  5  qt. 

5.  From  8  m.  130  r.  take  250  r.  3  yd.  2  ft. 

6.  From  a  hogshead  of  molasses  35  gal.  3  qt.  were 
drawn :  how  many  gallons  were  left  ? 


188  COMPOUND     SUBTRACTION. 

7.  If  one  farm  contains  165  A.  118  sq.  r.,  and  another 
100  A.  135  sq.  r.,  what  is  the  difference  between  them? 

8.  What  is  the  difference  between  two  loads  of  wood, 
one  of  which  contains  1  0.  38  cu.  ft.,  the  other  125 
cu.  ft.? 

9.  What  is  the  difference  in  the  weight  of  two  stacks 
of  hay,  one  of  which  contains  5  T.  135  lb.,  the  other 
7  T.  387  lb.? 

10.  The  longitude  of  New  York  is  740  o'  3*  W. ;  that 
of  Chicago,  870  35'  W.:  what  is  the  difference  in  their 
longitude  ? 

11.  The  latitude  of  New  Orleans  is  290  57'  30"  N«; 
that  of  Montreal  45°  31'  K:  what  is  the  difference  in 
their  latitude  ? 

12.  What  is  the  difference  of  time  between  Dec.  25th, 
1865,  and  April  20th,  1872  ? 

Solution.  —  Place  the  earlier  date 
under  the  later,  the  years  on  the  left, 
the  months  next,  and  the  days  on  the 
right,  and  proceed  as  in  subtracting  other 
Compound  Numbers. 

Remark. — In  finding  the  difference  between  two  dates,  and  in 
most  business  transactions,  30  days  are  considered  a  month,  and 
12  months  a  year. 

13.  If  a  man  was  born  Jan.  1st,  1850,  how  old  will  he 
be  July  4th,  1876  ? 

14.  A  note  dated  March  13th,  1870,  was  paid  Feb. 
25th,  1872  :  how  long  did  it  run  ? 

15.  Charles  was  born  July  30th,  1865,  and  his  brother 
Oct.  24-th,  1869  :  what  is  the  difference  in  their  ages  ? 

16.  A  whale-ship  started  on  a  voyage  Aug.  25th, 
1867,  and  returned  July  18th,  187 1,  how  long  was  she 
gone? 


Y. 

m. 

d. 

1872 

4 

20 

1865 

12 

25 

Ans 

.       6 

3 

25 

COMPOUND  MULTIPLICATION. 


i.  If  a  family  uses  2  lbs.  12  oz.  of  butter  a  day,  how 
much  will  they  use  in  3  days  ? 

Analysis. — They  will  use  3  times  2  lb.  and  12  oz.  Now  3  times 
2  lb.  are  6  lb.,  and  3  times  12  oz.  are  36  oz.,  equal  to  2  lb.  4  oz. ; 
which,  added  to  6  lb.,  make  8  lb.  4  oz.     Therefore,  etc. 

2.  If  it  takes  4  gal.  3  qt.  of  water  to  fill  a  demijohn, 
how  much  will  it  take  to  fill  2  of  the  same  size  ? 

3.  A  farmer  gave  a  bag  of  corn,  containing  2  bu.  3  pk.. 
to  each  of  4  beggars :  how  much  did  he  give  to  all  ? 

4.  If  it  takes  3  yd.  1  qr.  of  cloth  to  make  a  boy's  suit, 
how  many  yards  will  it  take  to  make  5  suits  ? 

5.  How  long  will  it  take  a  man  to  chop  3  cords  of 
wood,  if  he  chops  at  the  rate  of  a  cord  in  4  hr.  30  min.  ? 

6.  If  you  pick  2  qt.  1  pt.  of  blackberries  an  hour,  how 
many  can  you  pick  in  6  hours  ? 

7.  If  1  book  costs  2  shillings  and  6  pence,  what  will 
5  books  cost  ? 

SLATE    EXERCISES. 

1.  A  miller  ground  5  grists,  each  containing  2  bush- 
els, 3  pecks,  5  quarts  of  wheat :  how  much  wheat  did  he 
grind? 

ANALYSIS. — 5  grists  contain  5  times  as  much         Operation 
as  1  grist.     Now  5  times  5  qt.  are  25  qt.,  equal   '    Bu.     pk.     qt. 
to  3  pk.  and   1   qt.     Set    the   1  qt.  under  the         23c 
quarts,  and  carry  the  3  pk.  to  the  product  of 

pecks.     Next,  5  times  3  pk.  are  15  pk.,  and  3  are 1 

18  pk.,  equal  to  4  bu.  and  2  pk.     Set  the  2  under       14        2        1 
the  pecks,  and  carrying  the  4  bu.  to  the  product 
of  bu.  we  have  5  times  2  are  10  bu.,  and  4  are  14  bu.     Therefore, 
he  ground  14  bu.  2  pk.  1  qt. 


190  COMPOUND    MULTIPLICATION. 

3.  How  multiply  Compound  Numbers? 

I.  Write  the  multiplier  under  the  lowest  denomination 
of  the  multiplicand,  and  beginning  at  the  right,  multiply 
each  term  in  succession. 

II.  If  the  product  of  any  term  is  less  than  a  unit  of 
the  next  higher  denomination,  set  it  under  the  term  mul- 
tiplied. 

III.  If  equal  to  one  or  more  units  of  the  next  higher 
denomination,  carry  these  units  to  that  denomination, 
and  write  the  excess  under  the  term  multiplied. 


(2.) 

(30 

B. 

yd.      ft. 

M. 

fur.       r. 

yd. 

Mult. 

i3 

2          2 

30 

3       18 

4 

By 

5 

8 

Ans. 

67 

2           I 

243 

3       29 

4} 

4.  Bought  5  casks  of  vinegar,  each  containing  36  gal. 
3  qt.  1  pt. :  how  much  did  they  all  contain  ? 

5.  Sold  6  pieces  of  cloth,  each  containing  42  yards 
and  3  quarters :  how  much  did  all  contain  ? 

6.  A  farmer  has  4  pastures,  of  15  A.  63  sq.  r.  each : 
how  much  land  in  all  ? 

7.  A  man  bought  10  loads  of  wood,  each  containing 
1  C.  35  cu.  ft. :  how  much  wood  did  lie  buy? 

8.  If  you  read  5  h.  35  min.  per  day,  how  many  hours 
will  you  read  in  1 2  days  ? 

9.  Bought  7  loads  of  hay,  averaging  1  T.  375  lbs.: 
how  much  did  all  contain  ? 

10.  If  the  price  of  one  cow  is  £8,  15s.  6-Jd.,  what  will 
8  cows  cost,  at  the  same  rate  ? 

11.  If  you  have  11  apple-trees,  and  they  yield  7  bu. 
3  p"k.  apiece,  how  many  apples  will  you  have  ? 


COMPOUND     DIVISION. 


i.  If  48  lb.  12  oz.  of  rice  are  divided  equally  among 

15  persons,  what  part,  and  how  much,  will  each  receive  ? 

i 

!      Analysis. — 1  is  fV  of  15  ;  therefore,  each  per-  Operation. 

son  will  receive  1  fifteenth  part.  lb.      oz. 

Again,  1   fifteenth  of  48  lb.  is  3  lb.,  and  3  15)48      12 

remainder.    Reducing  the  remainder  3  lb.   to  . 

ounces,  they  become  48  oz.,  and  adding  the  12  * 

oz.  we  have  60  oz.    Now  1  fifteenth  of  00  oz.  is  4  oz.    Therefore, 
each  received  -fa  part,  which  is  3  lb.  4  oz. 

Note. — 1.  The  object  in  this  example  is  to  divide  a  compound 
number  into  equal  parts,  in  order  to  find  the  value  of  one  part. 

2.  A  farmer  sent  29  bu.  1  pk.  of  wheat  to  mill,  in  bags 
of  3  bu.  1  pk.  each :  how  many  bags  did  he  use  ? 

Analysis.  —  Reducing  the  whole  29  bu.  1  pk.=  H7  pk. 

quantity  to  pecks,  it  becomes  117  pk.  ,  l,u    j  r)k.—  j  -y  pk. 

The  quantity  in  each  bag,  3  bu.  1  pk.,  \tf* 

equal  13  pk.     We  now  divide  as  in  i_„JL 

simple  numbers.  AllS,    9  bags. 

Note. — 2.  The  object  of  this  example  is  to  find  how  many  times 
one  compound  number  is  contained  in  another. 

4.  How  divide  Compound  Numbers? 

I.  When  the  divisor  is  an  abstract  number, 
Beginning  at  the  left,  divide  each  denomination  in  suc- 
cession, and  set  the  quotient  under  the  term  divided. 

If  there  is  a  remainder,  reduce  it  to  the  n*x{  lower 
denomination,  and,  adding  it  to  the  given  up-tts  of 
that  denomination,  divide  as  before. 

II.  When  the  divisor  is  a  compound  number, 
Reduce  the  divisor  and  dividend  to  the  lowest  denom- 
ination contained  in  either,  and  divide  as  in  simple 
numbers. 


192  COMPOUND     DIVISION. 

Eemakk. — It  will  be  observed  from  the  preceding  examples, 
that  the  object  of  Compound  as  well  as  Simple  Division  is  twofold : 

ist,  To  divide  a  compound  number  into  equal  parts,  the  divisor 
being  abstract.  In  this  case  the  quotient  is  the  same  denomination 
as  the  dividend. 

2d,  To  find  now  many  times  one  compound  number  is  contained 
in  another.  In  this  case  the  quotient  is  times,  or  an  abstract  num- 
ber.   (P.  63,  Q.  10.) 

Perform  the  following  divisions : 

(3-)  (5-) 

5 )  16  A.  75  sq.  r.  35  sq.  ft.  2  lb.  4  oz. )  17  lb.  6  oz. 

(4.)  (6.) 

6)505  cu.  ft.  154  cu.  in.  £2,  i2S.)£23,  8s. 

7.  If  I  sell  46  bu.  1  pk.  of  plums  in  equal  quantities 
to  7  market-men,  how  many  will  each  receive  ? 

8.  Charles  having  a  kite-line  72  ft.  4  in.  long,  cut  it 
into  7  equal  parts :  what  was  the  length  of  each  part  ? 

9.  If  a  man  travels  48  m.  3  fur.  in  9  hours,  how  far 
will  he  go  in  1  hour  ? 

10.  A  goldsmith  made  5  lb.  3  oz.  of  silver  into  24 
spoons  :  what  was  the  weight  of  each  ? 

11.  How  many  iron  rails  18  ft.  long  are  required  to 
lay  both  sides  of  a  track  7  m.  160  r.  in  length  ? 

12.  A  man  gathered  57  bu.  3  pk.  of  oranges  from  9 
trees :  what  was  the  average  yield  ? 

13.  If  8  men  mow  17  A.  32  sq.  r.  in  a  day,  how  much 
can  1  man  mow  ? 

14.  How  many  times  does  a  car-wheel  16  ft.  6  in.  in 
circumference  turn  around  in  going  2  miles  ? 

15.  How  many  bags,  holding  2  bu.  3  pk.  each,  can  be 
filled  from  a  bin  which  contains  19  bu.  1  pk.  of  corn  ? 

16.  How  many  bundles  of  hay,  each  weighing  465 
pounds,  can  be  made  from  a  scaffold  which  contains 
5  tons,  125  pounds? 


PERCENTAGE 


1.  What  is  meant  by  per  cent  ? 
Per  cent  denotes  hundredths. 

Note. — The  term  is  from  the  Latin  per  and  centum,  by  or  in 
a  hundred. 

2.  What  is  the  rate  ? 

The  Mate  is  the  number  which  shows  how  many 
hundredths  are  taken.  Thus  i  per  cent  of  a  number 
is  i  hundredth  part  of  that  number  ;  2  per  cent,  2  hun- 
dredths ;  3  per  cent,  3  hundredths,  &c. 

3.  With  what  do  the  terms  rate  per  cent,  correspond  ? 

The  terms  Mate  per  cent  correspond  with  the 
terms  of  a  fraction,  the  denominator  of  which  is  always 
100,  and  the  numerator  the  given  rate. 

4.  How  then  may  per  cent  be  expressed  ? 

By  a  Common  or  a  Decimal  Fraction. 

5.  How  is  per  cent  expressed  by  decimals  ? 

Write  the  figures  denoting  the  per  cent  in  the  first  two 
places  on  the  right  of  the  decimal  point,  and  the  parts  of 
1  per  cent  in  the  succeeding  places  toward  the  right. 


1  per  cent  is 

written 

.OI 

\  per  cent  is  written 

.005 

6  per  cent 

a 

.06 

\  per  cent        " 

.0025 

10  per  cent 

a 

.IO 

2\  per  cent        " 

.025 

100  per  cent 

tt 

1.00 

6  J  per  cent        * 

.0625 

106  per  cent 

<( 

1.06 

33t  Per  cent        " 

•33l 

125  per  cent 

a 

1.25 

I07i  Per  cen^      " 

i-°75 

€-.  How  many  figures  are  required  to  express  per  cent? 
Per  cent  denotes  himdredths  ;  therefore  every  per  cent 
requires  at  least  two  figures,  to  express  it  decimally. 


194  PEKCENTAGE. 

7.  If  the  given  per  cent  is  less  than  10,  what  must  be  done  ? 
A  cipher  must  be  prefixed  to  the  figure  denoting  it. 
Thus,  i  per  cent  is  written  .01  ;  3  per  cent.  .03,  etc. 

Note. — When  a  given  part  of  1  per  cent  cannot  he  exactly 
expressed  by  one  or  two  decimal  figures,  it  is  written  as  a  common 
fraction,  and  annexed  to  the  figures  expressing  hundredths,  or 
the  per  cent.     Thus,  &%  is  written  .04^,  instead  of  .043333  +  . 

§.  To  what  is  100  per  cent  of  a  number  equal  ? 

A  hundred  per  cent  of  a  number  is  equal  to  the 
number  itself;  for  $%%  is  equal  to  1. 

9.  When  the  rate  is  100  per  cent  or  over,  how  is  it  expressed  ? 

By  a  mixed  number,  or  by  an  improper  fraction. 
Thus  125%  is  written  1.25,  or  -ffj. 

10.  What  is  the  sign  of  per  cent  ? 

The  Sign  of  per  cent  is  an  oblique  line  between 
two  ciphers  (%).    Thus,  2%  is  read  2  per  cent,  etc. 
"Write  the  following  per  cents  decimally : 

1.  4%;  7r/c;  10%;  45%;  103%;  110%;  205%. 

2.  2|%;  6-V/o;  7-|%;  i8f%;  io6J%;  iii{£ 

1 1 .  How  read  a  given  per  cent  expressed  decimally  ? 

Bead  the  first  Uvo  decimal  figures  as  per  cent,  and 
those  on  the  right  as  decimal  parts  of  1  per  cent. 
Copy  and  read  the  following  as  rates  per  cent : 

1.  .03;  .06;  .045;  .11J;  .625;  1.25;  1.50;  2.00. 

2.  1.06;  1.07;  1.08;  1.10^ ;  1.62-I;  1.005;  2.00. 

12.  What  are  the  elements  or  parts  in  calculating  percentage  ? 

The  base,  the  rate  per  cent,  the  percentage,  and  the 
amount. 

13.  Explain  each. 

The  Base  is  the  number  on  which  the  percentage  is 
calculated. 

The  Hate  per  cent  is  the  number  which  shows 
how  many  hundredths  of  the  base  are  to  be  taken. 


PERCENTAGE.  195 

The  Percentage  is  the  number  obtained  by  taking 
that  portion  of  the  base  indicated  by  the  rate  per  cent. 

The  Amount  is  the  base,  increased  or  diminished 
by  the  percentage. 

Note. — The  conditions  of  the  question  show  whether  the 
percentage  is  to  be  added  to,  or  subtracted  from  the  base  to  form 
the  amount. 

CASE    I 
To  find  the  Percentage,  the  Base  and  Rate  being  given. 

MENTAL    EXERCISES. 
i.  How  many  are  f  of  40  ?     (P.  90.) 
Analysis. — 1  fifth  of  40  is  8,  and  3  fifths  are  3  times  8,  or  24. 

2.  How  many  are  40  multiplied  by  |  ?    Ana.  24. 

3.  To  how  many  hundredths  is  f  equal  ? 

Analysis. — 1  =  \%% ;  hence  \  equals  \  of  |g&,  or  -fife ;  and  3 
fifths  =  3  times  -£&,  or  -&%-.    (P.  95,  Q.  20.) 

4.  What  is  T6o°{j-  of  40  ?    Ans.  24.     (P.  90.) 

5.  To  what  per  cent  is  ■££$  equal  ?    Ans.  60  per  cent. 

6.  What  is  60  per  cent  of  40  ? 

Analysis. — 60%  is  60  times  1  % .  Now,  1  %  of  40  is  -fa  of  40* 
or  YVo,  and  60  times  -,*„%  =  *&&t  or  24>  Ans. 

7.  To  what  per  cent  is  -j-jfo  equal  ?    yj^?    A0t>  ?    r&? 

8.  Eeduce  J  to  hundredths,  f  to  hundredths.  ^ 
to  hundredths. 

9.  To  what  per  cent  of  a  number  is  -J-  of  it  equal  ? 
Analysis. — £  equals  -ftftj ;  therefore,  £  of  a  number  is  20%. 

(P.  95,  Q-  20.) 

10.  To  what  per  cent  is  i  equal  ?    J  ?    -^  ?    ^  ? 

11.  What  is  5  per  cent  of  200  yards? 

Analysis. — $%  is  the  same  as  juu-  Now,  ffa  of  200  yards  is 
2  yards,  and  5  hundredths  are  5  times  2,  or  10  yards.  Therefore, 
$%  of  200  yards  is  10  yards. 

12.  What  is  7  %  of  $300  ?    8%  of  500  barrels  ? 


196  PERCENTAGE. 

13.  Which  is  the  greatest,  $  of  200;  or  200  x  •$;  or 
«oo  x  .60;  or  60  per  cent  of  200  ? 

SLATE    EXERCISES. 

Remakk. — From  the  preceding  illustrations,  it  will  be  seen 
that  finding  a  fractional  part  of  a  number,  multiplying  it  by  a 
wmmon  or  a  decimal  fraction,  and  finding  a  per  cent  of  it,  are 
identical  in  principle.  With  the  first  two  the  learner  is  supposed 
to  be  familiar ;  if  not,  he  should  carefully  review  them  before 
going  further.    (P.  90,  113,  139.) 

1.  What  is4#  of  $315  ? 

Analysis.— 4  per  cent  is  the  same  as  rfo*       operation. 
and  t^u  expressed  decimally  is  .04 ;  therefore,  *        g 

4  per  cent  of  $315  is  .04  times  $315.     Multiply-  -.     ' 

ing  the  base  by  the  rate,  expressed  decimally,  * 

we  have  $315  x  .04= $12.60,  the  percentage  re-  p 

quired. 

14.  How  find  the  Percentage,  when  the  base  and  rate  are  given? 
Etjle.  —  Multiply  the  base    by   the    rate,    expressed 

decimally. 

Notes. — 1.  When  the  rate  is  an  even  part  of  100,  the  per- 
centage may  be  found  by  taking  a  like  part  of  the  base.  Thus, 
for  20%,  take  £  ;  for  25^,  take  \,  etc.     (Ex.  3.) 

2.  The  amount  is  found  by  adding  the  percentage  to,  or  sub- 
tracting it  from  the  base,  as  the  case  may  be.     (Ex.  10.) 

2.  What  is  6%  of  $415.50?  Ans.  $24.93. 

3.  What  is  25  %  of  460  pounds  ? 
Solution. — 460  pounds  x  \  =  115  pounds,  Ans. 

4.  s?o  of  640  yards.  7.  8%  of  1000  rods. 

5.  6%  of  $765.60.  8.  12 #  of  1 1 10  barrels. 

6.  yfo  of  600  bushels.        9.  20%  of  2040  men. 

10.  A  farmer  having  163  acres  of  land,  bought  12  fc 
more,  how  many  acres  did  he  then  own  ? 

Solution.— 163  A.  x  .12  =  19.5C  A.  bought;  163  A.  +  19.56 
A.  =  182.56  A.  owned. 


PERCENTAGE.  197 

n.  A  man  having  560  sheep,  lost  2\%  of  them  by 
sickness :  how  many  did  he  have  left  ? 

Ans.— 5<5o  s.  x  .02^  =  14  s.  lost ;  560  s.  —  14  s.  =  546  s.  left. 

12.  A  teacher's  salary  is  12%  more  this  year  than  last; 
it  was  then  $1500  :  what  is  it  now  ? 

13.  From  a  school  of  750  pupils,  20  per  cent  were 
absent :  how  many  were  present  ? 

14.  What  is  62}  %  of  $25000  ? 


CASE    II. 
To  find  the  Hate,  the  Base  and  Percentage  being  given. 
1.  What  part  of  4  is  3  ? 

Analysis. — 1  is  \  of  4,  and  3  must  be  3  times  1  fourth,  or  3 
fourths  of  4. 

?.  What  part  of  5  is  3  ?    What  part  is  4  ? 

3.  To  how  many  hundredths  is  f  equal  ? 

4.  What  part  of  100  is  4  ?     What  part  is  5  ?     7  ?     9  ? 

5.  What  per  cent  of  a  number  is  y^  ?    T£g-  ?    -f^  ? 

6.  What  per  cent  of  $5  are  $2  ? 

Analysis.— $2  are  }  of  $5  ;  and  £  equals  ^0%  ;  therefore,  $2 
are  40%  of  $5.    (P.  95,  Q.  20.) 

7.  What  per  cent  of  10  yards  are  3  yards?    5  yds.? 

',yds.? 

8.  What  part  of  a  number  is  20%,  expressed  in  the 
lowest  terms  of  a  common  fraction  ? 

Analysis.— 20  per  cent  =  -ftfr  ;  and  -flfo  =a  \  Ans.    (P.  95.) 

9.  What  part  of  a  number  is  25%,  expressed  by  deci- 
mals? 

10.  What  part  of  a  number  is  5  per  cent?     10  per 
cent  ?    40  per  cent  ? 


198  PERCENTAGE. 


SLATE    EXERCISES. 

Remark. — From  the  preceding  illustrations,  it  will  be  seen 
that  finding  the  rate,  when  the  base  and  percentage  are  given,  is 
the  same  in  principle  as  finding  what  part  one  number  is  of 
another  ;  then  changing  the  common  fraction  to  hundredths. 

i.  A  man  paid  $28  for  a  cow,  and  sold  it  so  as  to  make 
$7  :  what  per  cent  did  he  make  ? 

Analysis. — In  this  example  the  base  $28,        operation. 
and  the  percentage  $7,  are  given,  to  find  the  28  )  7.00 

rate  per  cent.    Now,  $7  are  -^8-  of  $28  ;  and  

■fa  =  7  -*-  28  =  .25  or  25%.  Ans.  25% 

Or  fa  =  i  =  flfo  or  25  % .    (P.  95,  Q-  20.) 

15.  How  find  the  Rate,  when  the  Base  and  Percentage  are 
given? 

Divide  the  percentage  by  the  base  ;  the  first  tivo  deci- 
mal figures  will  be  hundredths,  or  the  rate  per  cent  j  the 
others,  parts  of  one  per  cent 

Note. — The  number  denoting  the  base  is  always  preceded  by 
the  word  of,  which  distinguishes  it  from  the  percentage. 

2.  A  lady  having  $40,  spent  $7  for  a  collar :  what  per 
cent  of  her  money  did  she  spend  ?  Ans.  174%. 

3.  What  %  of  18  is  6  ?  8.  What  %  of  135  is  4-5  ? 

4.  What  %  of  £1  are  £3  ?      9.  What  %  of  150  is  30? 

5.  What  %  of  $35  are  $7  ?     10.  What  %  of  J  is  J  ? 

6.  What  %  of  54  is  6  ?        11.  What  %  of  .8  is  .2  ? 

7.  What%  of  150 is  75?       12.  What  %  of  f  is  A? 

13.  A  man  bought  a  farm  of  200  acres,  and  sold  50 
acres  of  it :  what  per  cent  of  his  farm  did  he  sell  ? 

14.  A  farmer  having  500  sheep,  sold  125  of  them: 
what  per  cent  of  his  flock  did  he  sell  ? 

15.  If  a  man  earns  $450  a  year,  and  lays  up  $225  of 
it,  what  per  cent  of  his  earnings  does  he  save  ? 

16.  From  a  school  of  750  pupils,  250  were  absent: 
what  per  cent  were  absent? 


COMMISSION.  199 

APPLICATIONS    OF    PERCENTAGE. 

J.  To  what  classes  of  problems  is  percentage  applied  ? 

First,  To  those  in  which  time  is  one  of  the  elements 
uf  calculation  ;  as  Interest,  etc. 

Second,  To  those  which  are  independent  of  time  ;  a? 
Commission,  Profit  and  Loss,  etc. 

COMMISSION. 

2.  What  is  Commission,  and  how  computed  ? 

Commission  is  an  allowance  made  to  Agents,  Collec- 
tors, etc.,  for  the  transaction  of  business,  and  is  com- 
puted like  percentage. 

Notes. — i.  An  Agent  is  one  who  transacts  business  for  an- 
other, and  is  often  called  a  Commission  Merchant. 

2.  A  Collector  is  one  who  collects  debts,  taxes,  duties,  etc. 

3.  Goods  sent  to  an  agent  to  sell,  are  called  a  consignment ; 
the  person  to  whom  they  are  sent,  the  Consignee  ;  and  the  person 
sending  them,  the  Consignor. 

3.  What  answers  to  the  base,  the  rate,  etc.  ? 
The  amount  of  sales,  etc.,  is  the  base. 
The  per  cent  for  services,  the  rate. 
The  commission,  the  percentage. 
The  amount  of  sales,  etc.,  plus  or  minus  the  commis- 
sion, the  amount. 

To  find  the  Commission,  the  Amount  of  Sales  and  the 
Rate  being  given. 

1.  A  merchant  sold  goods  to  the  amount  of  $250,  at 
3  %  commission  :  what  was  his  commission  ? 

Analysis. — In  this  example  the  amount  of         Operation. 
sales  $250,  is  the  base,  and  3%  the  rate.     The  $250  B. 

question  then  is,  what  is  3%  of  $250.     Now,  #0,  J^ 

3%  equals  .03  expressed  decimally ;  therefore,  

3%  of  $250  is  .03  times  $250,  and  $250  x  .03   Ans.  $7.50  C. 
3=  $7  50,  the  commission  required. 


200  COMMISSION. 

2.  A  broker  sold  3  shares  of  bank  stock  for  $300: 
what  was  his  commission  at  -J  per  cent  ? 

Solution. — ^%  —  .005,  and  $300  x  .005  =  $1.50,  Ans. 

4.  How  find  the  Commission,  when  the  amount  of  sales  and 
rate  are  ^iven  ? 

Multiply  the  amount  of  sales  by  the  rate,  expressed 
decimally.    (P.  196,  Q.  14.) 

Remakes. — 1.  The  net  proceeds  of  a  business  transaction,  are 
the  gross  amount  Of  sales,  etc.,  minus  the  commission  and  other 
charges. 

2.  When  the  amount  of  sales,  etc.,  and  the  commission  are 
known,  the  net  proceeds  are  found  by  subtracting  the  commission 
from  the  amount  of  sales.    (Ex.  5.)    Conversely, 

3.  When  the  net  proceeds  and  commission  are  known,  the 
amount  of  sales  is  found  by  adding  the  commission  to  the  net 
proceeds.    (Ex.  6.) 

3.  A  man  collected  a  school  tax  of  $1250,  at  5^  com- 
mission :  how  much  did  he  receive  ? 

4.  If  you  sell  a  consignment  of  goods  for  $1175.60, 
what  will  be  your  commission  at  4%  ? 

5.  My  agent  sold  a  quantity  of  flour  for  $1585,  and 
charged  me  4$  :  what  was  Irs  commission,  and  what 
the  net  proceeds  ? 

Analysis.  — $1585  x  .04  =  $6340  com. ;  $1585  —  $63.40  = 
$1521.60,  net  proceeds. 

6.  Received  $115.20  for  selling  a  consignment  of 
goods,  and  sent  the  consignor  $2444.80  :  what  was  the 
amount  of  sales,  and  what  %  was  my  commission  ? 

Analysis.— $2444.804-  $115.20  —  $2560  sulos ;  ti^.co  -*-  2560  ■ 
,o$\%  commission. 

7.  My  agent  bought  28  shares  of  !N".  Y.  Central  R.  R. 
for  $2800,  and  charged  me  ij  per  cerA  ivouuivaioii: 
wfrat  was  his  commission  ? 


PROFIT      AND      LOSS,  £01 

PROFIT    AND    LOSS. 

5.  "What  are  Profit  and  Loss,  and  how  computed  ? 

Profit  and  Loss  are  sums  gained  or  lost  in  busi- 
ness transactions,  and  are  computed  like  Percentage. 

6.  What  answers  to  the  base,  the  rate,  etc.  ? 
The  Cost  or  sum  invested  is  the  Base  ; 
The  Per  cent  profit  or  loss,  the  Rate  ; 
The  Profit  or  Loss,  the  Percentage  ; 

The  Selling  Price,  that  is,  the  cost  plus  or  minus  the 
profit  or  loss,  the  Amount. 

To  find  the  Profit  or  Loss,  the  Cost  and  the  Rate  per  cent 
Profit  or  Loss  being  given. 

i.  Bought  a  horse  for  $150,  and  sold  it  for  12  per 
cent  profit :  how  much  did  I  gain  by  the  transaction  ? 

Analysis. — In  this  case  the  cost  $150  is  the  $150  B. 

base,  and    12%    the  rate.     Now,  12%   is  the  12  R. 

same  as  .12,  and  .12  times  $150  ss  $150  x  .12  =  . 

$18.00.    Therefore  I  gained  $18.  Am.  $18.00  €L 

2.  Bought  a  carriage  for  $250,  and  sold  it  at  a  loss  of 
8?c  :  how  much  did  I  lose  ? 

Analysis. — Here  the  cost  $250  is  the  base,  $2^0  B. 

and  8%   the  rate.     Now,  8%  is  the  same  as  08  E 

.08,  and  .08  times  $250  =  $250  x  .08  =  $20.00.  . 

Therefore,  my  loss  was  $20.  Ans.  $20.00  L. 

7.  How  find  the  Profit  or  Loss,  when  the  cost  and  rate  are  given  ? 

Multiply  the  cost  ly  the  rate  expressed  decimally,  as  in 
'percentage.     (P.  196,  Q.  14.) 

Remarks.— 1.  When  the  'per  cent  is  an  even  part  of  100  it  is 
generally  shorter,  and  therefore  preferable  to  nse  the  fraction. 

2.  The  selling  price  is  found  by  adding  the  profit  to  or  subtracts 
ing  the  loss  from  the  cost,  as  the  case  may  be.    (Ex.  9.) 


202  PROFIT      AND      LOSS. 

3.  A  man  bought  a  cow  for  $35,  and  sold  it  at  20% 
profit :  how  much  did  he  gain  ? 

4.  Henry  bought  a  pair  of  skates  for  $3.75,  and  sold 
them  for  20%  more  than  he  gave  :  what  was  his  gain  ? 

5.  A  grocer  bought  flour  at  $8.50  a  barrel,  and  sold  it 
at  25  %  loss  :  what  did  he  lose  on  a  barrel  ? 

6.  A  merchant  bought  a  piece  of  silk  for  $585,  and, 
sold  it  for  18%  advance  :  what  was  his  gain  ? 

7.  Bought  a  house  lot  for  $230,  and  sold  it  at  8%  less 
than  cost :  what  was  my  loss  ? 

8.  A  man  paid  $260  for  a  buggy,  and  sold  it  at  25% 
profit :  how  much  did  he  make  by  the  operation  ? 

9.  If  a  man  pays  $150  for  a  watch,  for  what  must  he 
sell  it  to  gain  12%? 

Analysis. — To  gain  12%  he  must  sell  it  for  the  cost,  plus  11%. 
Now,  12%  of  $150  =  $150  x  .12  ==  $18.00;  and  $150  +  $18  = 
$168,  Ana. 

10.  A  grocer  paid  $200  for  a  lot  of  peaches,  and 
finding  them  damaged,  sold  them  at  a  loss  of  25%  :  for 
what  did  he  sell  them  ? 

Analysis. — To  lose  25%  he  must  sell  them  for  the  cost,  minus 
25%.  Now,  25%  of  $200  is  $50;  and  $200  —  $50  =  $150,  the 
selling  price. 

11.  Bought  a  case  of  12  hats  at  $6.50  apiece:  for 
what  must  I  sell  the  whole  to  make  20%  ? 

12.  A  dealer  bought  50  buffalo  robes,  at  $8  apiece, 
and  sold  them  at  16%  loss  :  what  did  he  get  for  them  ? 

13.  A  merchant  bought  240  barrels  of  flour,  at  $7.50 
a  barrel,  and  sold  it  at  a  profit  of  1 2  J  %  :  what  did  the 
flour  come  to  ? 

14.  Bought  a  farm  for  $4200,.  and  sold  it  for  23$  % 
more  than  cost :  for  how  much  was  it  sold  ? 


PROFIT      AND      LOSS.  203 

To  find  the  per  cent  Profit  or  Loss,  the  cost  and  the  amount 
of  Profit  or  Loss  being  given. 

i.  I  bought  a  sleigh  for  $50,  and  sold  it  for  $20  more 
than  the  cost :  what  per  cent  was  the  profit  ? 

Analysis. — The  gain  $20  is  H  of  $50,  the  cost ;  now  §  %  = 
tW  or  40  per  cent.  (P.  198,  Q.  15.)  Therefore  the  profit  was 
40  per  cent. 

8.  How  find  the  rate  per  cent,  the  cost  and  the  amount  of  profit 
or  loss  being  given  ? 

Divide  the  amount  of  profit  or  loss  by  the  cost,  as  in 
percentage.    (P.  198,  Q.  15.) 

Remark.  —  When  the  cost  and  selling  price  are  given,  the 
Profit  is  found  by  subtracting  the  cost  from  the  selling  price.  The 
Loss  is  found  by  subtracting  the  selling  price  from  the  cost. 

2.  Bought  a  horse  for  $200,  and  sold  it  for  $50  less 
than  cost :  what  per  cent  was  the  loss  ? 

3.  A  fruit  dealer  bought  oranges  at  4  cents,  and  sold 
them  so  as  to  make  2  cents  on  each  :  what  per  cent  was 
his  profit  ? 

4.  If  you  buy  a  slate  for  10  cents,  and  sell  it  for  5 
cents  more  than  cost :  what  per  cent  is  your  gain  ? 

5.  Paid  $35  for  a  cow,  and  sold  her  for  $10  less  than 
cost :  what  per  cent  was  the  loss  ? 

6.  If  a  man  buys  flour  at  $7.50  a  barrel,  and  sells  it 
for  $8.50,  what  per  cent  is  his  profit  ? 

7.  If  you  buy  tea  at  60  cents  a  pound,  and  sell  it  at 
80  cents,  what  per  cent  is  your  profit  ? 

8.  If  a  grocer  buys  sugar  at  8  cents  a  pound,  and  sells 
it  at  6  cents,  what  per  cent  is  his  loss  ? 

9.  A  fruit  dealer  bought  bananas  at  $3.50  a  hundred, 
and  gpld  them  at  $5  a  hundred :  what  per  cent  did  he 
male? 


INTEREST. 


1.  What  is  interest? 

Interest  is  a  compensation  for  the  use  of  money. 

,      2.  What  are  the  elements  or  parts  to  be  considered  ? 

The  Principal,  the  Rate,  the  Interest,  the  Time,  and 
the  Amount. 
3.  Explain  each. 

The  Principal  is  the  money  lent. 
The  Mate  is  the  per  centner  annum. 
The  Interest  is  the  percentage. 
The  Time  is  the  period  for  which  the  principal  draws 
interest. 
The  Amount  is  the  sum  of  the  principal  and  interest. 

Note. — The  term  per  annum,  from  the  Latin  per  and  annus, 
signifies  by  the  year. 

To  find  the  interest  of  $1,  at  6  pep  cent  for  months. 

Remark. — The  learner  should  observe  that  Interest  differs 
from  the  preceding  applications  of  Percentage  by  introducing 
time  as  an  element  in  connection  with  the  rate  per  cent.  The 
terms  rate  and  rate  per  cent  always  mean  a  certain  number  of 
hundredths  yearly,  and  pro  rata  for  longer  or  shorter  periods. 

i.  If  I  charge  6%  yearly  for  the  use  of  $i,  how  much 
'shall  I  receive  ? 

Analysis. — 6  per  cent  is  Tott,  and  $i  is  ioo  cents.  Now,  Tfj  j 
i>f  ioo  cents  is  6  cents.     Therefore,  I  shall  receive  6  cents. 

2.  If  the  interest  of  $i  for  i  year  is  6  cents,  how  much 
is  it  for  i  month  ?     For  2  months  ?     For  3  months  ? 

Analysis.— Since  the  interest  of  $1  at  6  per  cent  for  12  months 
(1  year)  is  6  cents,  for  1  month  it  is  1  twelfth  of  6  cents,  and  f§ 
of  6  cents  is  -ft  or  \  cent.  Again,  since  the  interest  of  $1  at  6% 
Is  \  cent  for  1  month ;  for  2  months  it  is  2  halves  or  1  cent ;  for  3 
months,  3  halves  or  li  cent ;  for  6  mos.,  6  halves  or  3  cents,  etc. 


INTEREST.  205 

4.  What  then  is  the  interest  of  $i  at  6  per  cent,  for  any  number 
of  months  ? 

The  interest  of  SI,  at  6  per  cent,  for  any  number  of 
months,  is  half  as  many  cents  as  months. 

4.  What  is  the  interest  of  $1,  at  6%  for  5  months  ? 

5.  What  is  the  interest  of  $1,  at  6fo  for  7  months? 

To  find  the  interest  of  $1,  at  6%  for  days. 

6.  What  is  the  interest  of  Si,  at  6%  for  1  d.  ?  For  2 
d. ?    3d.?    4  d. ?    5  d. ?    6  d. ?  etc. 

Analysis. — Since  the  interest  of  $1  for  30  days  (1  month),  is 
\  cent,  or  5  mills,  for  1  day  it  is  1  thirtieth  of  5  mills,  and  -3Vr  of  5 
is  -nr,  or  £  mill.  Again,  since  the  interest  of  $1  for  1  day  is  £  mill, 
for  2  days,  it  is  §  mill ;  for  3  days,  %  ;  for  6  days,  f ,  or  1  mill. 

5.  What  then  is  the  interest  of  $i,at  6  per  cent  for  any  num- 
ber of  days  ? 

The  interest  of  SI,  at  6  per  cent,  for  any  number  of 
days,  is  1  sixth  as  many  mills  as  days. 

7.  What  is  the  interest  of  $1,  at  6%  for  19   days? 

8.  What  is  the  interest  of  $1,  at  6  ft  for  15  days  ?  For 
23  d.?     25  d.  ? 

To  find  the  interest  of  $f,  at  6  per  cent  for  months  and  day9. 

9.  What  is  the  interest  of  $1,  at  6fo  for  4  m.  21  d.  ? 
Analysis. — The  interest  of  $1  for  4  m.  is  £  of  4  or  2  cents  ; 

and  the  interest  of  $1  for  21  d.  is  £  of  21  or  ?>\  mills,  which,  added 
to  2  cts.,  make  2  cts.  +  3^  mills,  or  $.0235. 

6.  How  find  the  interest  of  $1,  at  6fc  for  months  and  days. 
Take  half  the  number  of  months  for  cents,  and  one 

sixth  the  number  of  days  for  mills.     TJie  sum  will  be  the 
interest. 

Note. — In  finding  1-sixth  of  the  days,  it  is  commonly  sufficient 
to  carry  the  quotient  to  tenths  or  hundredths  of  a  mill. 

10.  What  is  the  interest  of  $1,  at  6%  for  6  m.  24  d.  ? 

11.  What  is  the  interest  of  $1,  at  6%  for  8  m.  27  d.  ? 


206  INTEREST. 


General  Method  of  Computing  Interest,  the  Principal,  the 
Time,  and  the  Rate  being  given. 

12.  What  is  the  interest  of  $115.20  for  1  y.im.  and 
12  d.,at6#? 

Analysis.  —  1  year  equals  12  m.,  Operation. 

and  12  m.  +  1  m.  -        -        =13111.  $1 15.20 

The  int.  of  $1  for  13  months  -        =  $.065  0fij 

"  "         12  days      -     .  -        ss    .002         

The  int.  of  $1  for  iy.im.  and  12  d.     =  $.067  $80640 

Since  the  int.  of  $1  for  the  given  time  is  69120 


$.067,  or  .067  times  the  principal,  the  int.  of       a         o 
$115.20  must    be   .067  times  that  sum,  and         '*' 
$115.20  x  .067  =  $7.7184,  Afl8. 

13.  What  is  the  interest  of  $150  for  10  months,  at  7 
per  cent  ? 

Analysis. — At  6%  the  int.  of  $150  for  the 
time  is  $150  x  .05  =$7.50.  But  7%  is  1%, 
or  £  more  than  b%  :  and  $  of  $7.50  is  $1.25. 
Now.  $7.50  +  $1.25  =  $8.75,  the  interest  at  7 
per  cent. 


Ans.  $8.75 

9*.  How  find  the  interest  on  a  given  principal,  for  any  given 
time  and  rate  ? 

I.  When  the  rate  is  6  per  cent, 

Multiply  the  principal  by  the  interest  of  $1,  at  6  per 
wit  for  the  time,  expressed  decimally. 

II.  When  the  rate  is  greater  or  less  than  6  per  cent, 
Add  to  or  subtract  from  the  interest  at  6  yer  cent  such 

a  part  of  itself  as  the  given  rate  exceeds  or  falls  short  of 
6  per  cent. 

§.  How  find  the  amount  ? 

Add  the  interest  to  the  principal. 

Notes. — 1.  In  finding  the  time,  first  determine  the  number  of 
entire  calendar  months ;  then  the  number  of  days  left. 


1KTEKEST.  207 

2.  In  computing  interest,  if  the  mills  are  5  or  more,  it  is 
customary  to  add  1  to  the  cents  ;  if  less  than  5,  they  are  dis- 
regarded. 

Only  three  decimal  figures  are  retained  in  the  following  answers : 

14.  Find  the  amount  of  $75.60  for  1  y.  3  m.  9  d.,  at 

6%. 

Solution. — Int.  of.$i  for  the  given  time  and  rate,  is  $.0765. 
Now,  $75.60  x  .0765  =  $5.7834,  the  int.  The  prin.  $.75.60  -V 
$57834  ==  $8l-383>  Amt. 

15.  Find  the  int.  of  $45.50  for  1  y.  7  m.  15  d.,  at  y%. 

16.  Find  the  amt.  of  $58.75  for  1  y.  10  m.  21  d.,  at  6#>* 

17.  The  int.  of  $85  for  8  m.  9  d.,  at  5%. 

18.  The  int.  of  $113  for  7  m.  18  d.,  at  6%. 

19.  The  int.  of  $150  for  1  y.  3  m.,  at  6%. 

20.  The  int.  of  $265  for  1  y.  7  m.,  at  6%. 

21.  The  amt.  of  $500  for  2  y.,  at  8%. 

22.  The  amt.  of  $763.25  for  1  y.  9  m.  27  d.,  at  5$. 

23.  The  int.  of  $1500  for  3  years,  at  2>%. 

24.  The  int.  of  $2678  for  1  y.  7  m.  19  d.,  at  7%. 

25.  The  int.  of  $2750  for  ^  days,  at  6%. 

26.  The  int.  of  $3700  for  6^  days,  at  7%. 

27.  The  int.  of  $2500.73  for  93  days,  at  5%. 

28.  What  is  the  int.  on  a  note  of  $500,  from  March 
10th,  1872,  to  July  25th,  1872,  at  6%  ? 

Analysis. — The  time  from  March  10th  to  July  10th,  is  4  m. ; 
from  July  10th  to  July  25th,  it  is  15  d.  Now,  $500  x  .0225  = 
$11.25,  Ans. 

29.  What  is  the  amt.  of  $1250,  from  July  20th,  1873, 
to  Dec.  29th,  1873,  at  6%  ? 

30.  What  is  the  amt.  of  a  note  of  $2000,  bearing  int. 
from  March  1st,  1872,  to  Jan.  25th,  1873,  at  7  %  ? 

31.  What  is  the  interest  of  $4500  for  five  years,  at  6%  ? 
The  amount  ? 


ANSWERS 


ADDITION. 


Ex. 


Ans. 


Page  21. 

2.  24 

3.  25      * 

4-  32 

5-  27 

6.  26 

7.  27 

8.  34 
9-  37 

1.  45 

2-  45 

3-  52 
4  59 

5.  60 

6.  61 
7.58 
8.  66 

Page  23. 

1.  47 

2.  41 

3-  44 

4-  55 

5-  53 

6.  62 

7.  64 

8.  70 

9.  22  pounds 
10.  35  yards 

Page  24. 

2.  67 

3-  88 

4-  79 
5.  98 


Ex. 


Ans. 


6.  97 

7.  887 

8.  888 

9.  999 

10.  67 

11.  77 
89 
95 


12. 

13- 
14. 

i5- 


99 

rage  28. 

13142 
16424 
16189 
17 140 

17374 
1714  yds. 

2453  lbs. 
2359  rods. 
$2263 

2454  A. 

Page  29. 

6.   3280 

7-  1936 

8.  1232 

9.  2093 

10.  2377 

11.  861  pages 

12.  2764  days 

13.  $369 

14.  $119 

15.  939  sheep 

16.  1682  sheep 


Ex. 


AN8. 


17.  27514 

18.  84OI8 

19.  1799  A.D. 

20.  $7835 

21.  1886  A.D. 

Page  30. 

22.  2387  yds. 

23.  $2123 
318  years 
208  mar. 
$2526 
45913  men 

28.  $58020 

29.  1492  days 

30.  705  miles 

31.  998  acres 

32.  11252  years 

33.  2413  oz. 

34.  2969  lbs. 

35.  $1700 

36.  1003  A. 


24. 

25- 
26. 

27. 


Page  31. 

37.  #5<>3 

$593 
$6674 

$53.68 

$48.57 
$62.55 

#539-°3 
$829.03 
$648.62 

I7M-57 

$6366.10 


38. 

39- 
40. 

41. 
42. 

43- 

44. 

45- 
46. 

47- 


ANSWERS, 


209 


SUBTRACTION. 


Ex.    Ans. 

Ex.     Ans. 

Ex.    Ans. 

Page  38. 

4.  2223 

17.  3240230 

2.  213 

5.  I409 

18.  1765509 

3-  324 

6.  1804 

19.  3929992 

4.  320 

20.  6706495 

5.  2232 

I-  235 

21.  2235IO9 

6.  4063 

2.  3108 

22.  IOOOOO4 

3-  3^ 

23.  582  sh. 

Page  39. 

4.  144 

24.  $422 

1.  223  lbs. 

5-  JI74 

25.  83  years 

2.  263  yds. 

6.  5218 

26.  $588 

3.  4316  hats 

7.  3101 

27.  64  years 

4.  $1111 

8.  2228 

28.  1722 

7.  3562  in. 

9.  2167  qts. 

29.  $930 

8.  5101  oz. 

10.  1447 1 7  bar. 

30.  156  years 

9.  3000  weeks 

10.  32 1 1 

Pagre  45. 

Page  44. 

11.  4000 

11.  738  bu. 

1.  504 

12.  $723 

2.  100 

Page  42. 

13.  $405 

3.  1409 

i.  Given 

14.  990 

4.  2640 

2.  2191 

15.  990 

5-  !5i64 

3-  2313 

16.  7272 

6.  4325 

MULTIPLICATION 


Page  50. 

2.  84828 

3.  66963 

4.  48848 

5-  55555 

6.  606606 

7.  996092 

8.  770707 

9.  888888 

Page  52. 

2.  1825  days 

3.  $4500 


4.  7000  A. 

5.  4272 

6.  22035 

7.  114144 

8.  325215 

9.  4207380 

10.  4450096 

11.  6643240 

12.  8757720 

Page  54. 

2.  93100 
1  3-  I39502 


4.  247388 
5-  344109 
6.  681252 

2/574476 

3.  1086912 

4.  1830048 

5.  3299541 

6.  99764360 

7.  126257940 

8.  296989550 
9-  M37399648 


210 

ANSWEBS. 

Ex.     Ans. 

Ex.     Ans. 

Ex.     Ans. 

io.  8760  hrs. 

31.  $655500 

2°-  33995°° 

11.  48000  rods 

32.  1 1 6300  p. 

21.  85442OOO 

12.  $23055 

33-  #59985 

22.  24139500 

13.  $24250 

34.  $188040 

23.  195500000 

14-  5393°58 

24.  21375000000 

l5-   18305988 

Page  57* 

25.  4186100000 

16.  143225262 

1-3.  Given 

26.  480480120000 

17.  168465500 

4.  36100 

27.  $14000 

18.  213882848 

5-  453°° 

28.  1260000  cts., 

19.  229152462 

6.  2045000 

or  $12600 

20.  186691875 

7.  46208000 

29.  36000  miles 

21.  411290946 

8.  58241000 

30.  150000  bu. 

9.  3260720000 

31.  62500000 

Page  56. 

10.  400728900000 

32.  3920000000 

22.  $55250 

11.  $510 

33.  1412019000 

23.  22770  miles 

12.  $26500 

34.  28860000000 

24.  16884  yards 

13.  $20500 

25.  $160000 

14.  63000  cts.,  or 

Page  59. 

26.  $19250 

$630 

1.  5130  mo. 

27.  $40250 

2.  56250  d. 

28.  $123375 

Page  58. 

3-  $576 

29.  $282750 

15-18.  Given 

4.  $460 

30.  $200000 

SF 

19.  1 147000 
[  O  R  T  D I  VI S I  0 

5.  $1464 

N. 

i*age  65. 

5.  I201lf 

11.  460  hats 

~   2131 

12.  587  barrels 

3.  2031 

Fage  68. 

13.  52-f  weeks 

4.  2101 

2.  1 1708 

14.  125  hrs. 

5.  IOIOI 

3-  11389! 

15.  52  boats 

6.  2021 

4.  12004 

16.  24  weeks 

7.  IIOI 

5.  18062 

17.  70  barrels 

8.  1010 

18.  170  boxes 

£.  IOIOI 

Page  60. 

19.  $6253 

6.  112363- 

20.  2808  boxes 

?age  67. 

7.  10715} 

21.  2203  A. 

2.  3H2 

8.  II202| 

22.  $10603 

j.   20143 

9.  1 0338  £ 

23.  2500  hours 

4  3121? 

10.  282  barrels 

24.  125  stages 

ANSWERS. 

211 

LONG     DIVISION. 

Ex 

Ans. 

Ex.          Ans. 

Ex.          Ans. 

Page  71. 

24.  60  days 

2.   90000  Cts., 

I. 

Given 

25.  $68Jf£ 

or  %oo 

2. 

Given 

26.  $28 

3.  $1605 

3- 

4- 

3476J- 
7275t 

Page  74, 

4.  $5 

5.  105^  tons 

1.  Given 

6.  $945 

Page  72. 

QTI9 

2.  85,  64  rem. 
3-  46,  53 T  rem. 

-P«*/e  78. 

i. 

4.  48 

7.  685  barrels; 

2. 

2  242f| 

5.  437, 5681  rem. 

$7  a  bar. 

3- 

2I75A 

6.  3,  9467  rem. 

8.  $2733 

4- 

I756HI 

7.  2,  72364  rem. 

9.  $2310 

5- 

I2  724H 

8.  10 

10.  $1305 

6. 

l6lo44 

9-  85>  325764r. 

«•  25435 

53 

10.  Given 

12.  29379 

7- 

I333Iff 

11.  426,  14  rem. 

13.  41884 

8. 

13379 

12.  411,  15  rem. 

14.  30  days 

9- 

10328IJ 

13-  85>  J545  rem. 

15.  4  years 

IO. 

99289H 

14.  78,  281  rem. 

16.  164  cts.,  2d 

ii. 

204  cows 

15.  31,  342  rem. 

311  cts.,  3d 

12. 

200-JI  A. 

16.  8,  16 168  rem. 

17.  25  days 

13- 
I4. 

36  months 

180  stoves 

Page  75. 

Page  81. 

15- 
16. 

i57ff  years 
83JJ  months 

1.  244 

2.  775 

1.  Given 

2.  2x3x7 

3.  2x2x2x2x3 

*7- 

18. 

99^  hhd. 
75  yoke 

Page  76. 

3.  1089 

4.  2x2x3x5 

5.  2x2x2x3x3 

6.  2x2x5x5 

Page  73. 

4.  2510 

5.  606 

6.  1747 

7.  6500 

7-  5X5X5 

19. 
20. 
21. 

Given 

i329*£ff 
10266^ 

8.  2x2x3x11 

9-  5x5x7 
10.  2x2x2x5x5 

II.2X2X2X2X? 

22. 

3oio|IM 

jPagre  77. 

X  2  X  2  X  2 

23- 

3°2  2y-6-2T 

1.  5  days 

12.    5X5X13 

212 

ANSWERS. 

Ex.         Ans. 

Ex.           Ans. 

Ex.           Ans. 

13-  2x5x5x3x3 

IO.    II 

14.   32 

14.  5X5X5X5 

11.  -*/,  or  i2|  yd. 

15.    12  ft. 

15.  2X2X2x5x5 

12.  36  barrels 

16.  20  yds. 

x5 

13.  44  days. 

16.  2x2x2x2x2 

X2X3X3X3 
fj,  2x2x2x2x3 

Page  85. 

Page  87. 
1.  Given 

X3X13 

1.  Given 

2.  24 

2.  9 

3-  48 

Page  82. 

3.  16 

4-  J5 

4.  90 

5.  1008 

1.  Given 

5-24 

6.  15 

7.  27 

6.  1800 

2.  Given 

7.  660 

3.  Given 

8.  156 

4.  Given 

8.  4 

9.  648 

5-  h  or  2  J 

9.  256 

10.  240 

6.  J£,  or  if 

10.  12 

11.  168 

7.  V>  or  3? 

11.  15 

12.  12960 

8.  4 

12.  15 

13-  864 

9.  14 

13.  48 

REDUCH 

:iON     OF    FRACTIONS. 

J'aflre  95. 

13.  i 

2.  Givem 

1.  Given 

14-  i 

3-1 

2.  « 

15.  f 

4-  I 

~      25 
3'    ^HT 

4-    « 

16.  } 

17.  i 

5-  i 
6.| 

5  t# 

18.21 

7-1 

6.  41 

9 

8.  1 

7.H 

19.  f 

9.  A 

8-  tV\ 

20.  -J 

10.  i 

9.  *& 

•i.2i 

11.  V 

"•   TOlfe 

12 

12.  -B 

i'.  Given 

Page  96. 

13.  * 

'  .* 

1.  Given 

14-  A 

ANSWERS. 


213 


Ex.              Ans. 

Ex.            ANS. 

Ex. 

Ans. 

i5-  U 

19.  $3i 

2. 

Given 

ik&t 

3- 

a 

17-  i 

Pagre  99. 

is.  a 

1.  Given 

JPttgre  J  00. 

19-  i 

2.  Given 

4- 

Given 

20.  J 

21.  H 

•  3-  ¥ 
4-  *¥■ 

5- 
6. 

Given 

* 

22.  tV* 

5-  H* 

7- 

Tl^f 

23-   A 

6.    ±fi 

8. 

* 

24.  i 

7.  ¥ 

9- 

j 

25.* 
26.il 

8.  ^ 

9-  -v- 

10. 
11. 

i 

10.  ip- 

12. 

i 

A 

H 

A 

rage  97. 

11. 4* 

13- 

1.  Given 

12.  ^ 
it.  •s-47 
14.5 

i5-  W 
16    xt§7 

14. 
*5- 

2.  19 
3-  »4l 

16. 
17- 

4-  16  J 

18. 

5 
T 

5-  i6| 

xu*         15 
T-       2003 

19. 

3 

6.  12 

*7«    —20- 

18.  a$|* 

19.  ^P 

20.  V 

21.  ^F 

22.  7  °  °  3 
„„      85  19 

20. 

ft 

7-   My 

8.  12A 

9.  15 

1.  Given 

10.  8 

2. 

1* 

".  5ft 

3- 

if 

12.  si 

13.  H 

14.  7f 

3*  — ?r — 

24.  a^ul 

25.     15233 

4. 

5- 
6. 

if 

51 

i5-  7fi 

26.  i^o-U. 

7- 

81 
T77 

16.  8 

27.  103  beg. 

8. 

87 
TTT 

17.  12 

9- 

225 

18.  2|sh. 

1.  Given 

10. 

500 
TWO" 

214 

ANSWEES 

1. 

Ex.             Ans. 

Ex. 

AN8. 

Ex.           Ans. 

Page  102. 

-        8 
3*    T2> 

9 

fi      36      40     45 
°-    o"T7>   6"T3T>  ^0" 

i.  Given 

/i        6         5 
4'   TT?  TT 

7-  18,  ft,  IS 

-       4  0      4  5      12 

5*    "2~8"> 

12 

2'    ZH>  3T»  ^TT 

1^ 

8      JUL      144    X85. 
°-    TT5>  ^T6>   2T# 

4- 

8        5        6 
~2Ut  T0~>  TO" 

«      315      616      297 
9*    3~9~T>  ^9~3>  ^9T 

5- 

21     10      4 
T4>  T4>  ?4~ 

T~      31080     47730      16340 
IO*    S~TSWUt  6  3  640?  6  364tf 

6. 

30     2  8     4  9 
TO?  T0>  TU 

XI«    J^OOO?  IfOOO?  TTTTTT 

7- 

Ti'5>  T5>  T6~ 

12.  Given 

8. 

TUT?  TOT?  TOT 

13-  n,  a,  f? 

9- 

TT>  TT>  IT 

i4.a%%1 

IO. 

n. 

4T>  if>  Jf  ?  A 

1320     1617      525       2695 
~3~4~6T>  T4~6T>  3  46  5>  TT6T 

Page  103 

• 

12. 

Given 

i,  2.  Given 

13- 

16     2  10     2  5 
4T(5>  ~4TT?  4~tf 

3*    3~0~>  3~0~>  317 

14. 

t >  ¥>  1.  ¥ 

ADDITION    OF    FRACTIONS 


Page  105, 

i.  Given 
2.  ^  =  2|  A. 
3-*!=  * 
4-  M  =  ^ft 

5- «=»* 

6.  «=i« 


Page  106. 

4-    H=H 

6-ft=i* 

7-  W=«* 


«•«  =  '« 
9-  W  =  'ft 

II.  «f  =  2T«& 

13.  ioif-lbs. 

14.  52-^  yd. 
i5-  77W  m- 

16.  $24J 

17.  l7i« 

18.  Given 

19.  5« 

20.  28J 

■I.  23I 

22.  46f} 


ANSWERS 


315 


SUBTRACTION     OF    FRACTIONS. 


Ex.           Ans. 

Ex.         Ans. 

Ex.       Ans. 

Jt-tef/d  107. 

3-T5- 

Pa#e  J  0,9. 

i,  2.  Given 

4-  A 

15.  Given 

*Hf&$ 

5-  A 

16.  2 if  bu. 

4.  U  bu. 

6-Ni 

17.  Given 

e      60 

7-  T% 

18.  35lgal. 

6-tWr 

8.  A 

19.  18J 

7- jib 

9.  ?£=$ 

20.  35  A 

8-T§*T 

i°-  u 

21.  20f 

TT      44 

22.  29J 

JPacre  20S. 

12.  A 

24.  A=* 

i.  Given 

13- A* 

25-A=J 

2-H 

*4-Ab 

MULTIPLICATION    OF    FRACTIONS. 


Page  111 

1. 

Given 

2. 

*6f 

3- 

$toA 

4- 

$i5°A 

5- 

»*A 

6. 

i5ii 

7- 

8H 

8. 

7t3t 

9- 

*H 

10. 

"A 

11. 

26J. 

12. 36  If 

13-  62J 

14.  42f  J 

15.  Given 


16.  $2.87} 

17.  $843 J 

18.  781J 

19-  1575 

20.  2478I 

21.  3562I 

22.  5000 

23.  8775 

24.  $654 

25.  $75° 

26.  $2224j 

1.  Given 

2.  45  cts. 
3-  $62^ 


4-  33i 

5-48f 

6.  22f 

7-  3*1 

8.30 

9-741 

10.  257A 

11.  135 

12.  292  days 

Page  114. 

13.  Given 

14.  $7 

15.  486  miles 

16.  325 
i7-44i| 


216 


ANSWERS, 


Ex.           Ans. 

Ex.              Ans. 

Ex.             Ans. 

i  8.  663 

8.| 

4-  Hfj 

19.  1090J 

9-1% 

5-  5°f 

20.  4161^ 

10.  A 

6.  14J 

21.  6250 

"•A 

I2-  A 

Jtogre  J 16. 

Page  115. 

13- A 

7-  Ii.6si 

1.  Giyen 

14.  A 

8.  240I 

2.f& 

15.  A 

9-  159ft 

3-$A 

16.  Given 

10.  585« 

4-1 

11.625 

5-1 

i-t4t 

12.  1431J 

M 

2.  J 

13-  2503H 

7-H 

3-f 

H.  6737M 

DIVISION     OF    FRACTIONS. 


Page  117. 

17. 3H 

Page  120. 

1,  2.  Given 

TQ     ,,293 

2.6 

3- A 

Tn     ?293 
19-  23^<J 

3-  4j 

4-H 

20.  4fli 

4.8f 

5- A 

5-4 

6.  A 

rage  119. 

Mtt 

7-t** 

1,  2.  Given 

7-  4i 

S-tH* 

3-53J 

9-  A 

4.88 

JPwflre  Jf£l. 

i°-  dflft 

5-  H3i 

2.  i|  lb. 

11.  A^ 

6.  192^ 

3.  2^  pines 

12.  Aft 

7-  5A 
Mtt 

4.  6}  lb. 

Paf/e  iiS. 

9-6f 

Page  122. 

14.  $7i 

10.  8f 

5-3} 

15-51 

11.  $4 

6.1* 

16.  3A 

12.  4  miles 

7-  iA 

ANSWERS, 


217 


Ex.     Ans. 

Ex.     Ans. 

Ex.     Ans. 

8.  2f 

18.  I» 

Page  123. 

9.  Iff 

19-  2^ 

4-  & 

10.5 

20.  2|| 

5-  ^T 

"*irtfr 

21-  3if 

6-i 

12.  2f| 

2  2.  2f| 

7.  A 

13-  2| 

23.2^ 

8.  A 

14-  2^6T 

25-5 

9-  *i* 

15-  2j 

26.  A 

10.  ^ 

*"•  X 228  25 

27.  60 

11.* 

QUESTIONS    FOR     REVIEW. 


Page  124. 

1.  to! 

2.  $I2| 

5-  *38« 

6.  37  J  lb. 

7.  96  p.  k. 

8.  1 46  J  lb. 

9.  4|  lb. 

10.  8  cords 

11.  5  bar. 
is. -It*  sum; 


TO? 


dif.; 
A>  prod.; 
3i  quot. 

13.  2  if  days 

14.  8  bales 
15-  Mi 

16.  $9 J 
i7-4ilb. 

18.  3f 

19.  $i7f 

20.  3500  lb. 


21. 


fjsold; 
U  left 


22.  45  weeks 

23.  hsi 

24.  $9A 

25.fi  yd. 

26.  A 

27.  35 

28.  i« 

29.  6f 

30.  $8,  flour 

31.  14  oranges 

32.  5  lb. 


FRACTIONAL 
rage  126. 

28  —  4 

1     27  — I  . 
3«  "ST  —  U 
6  3  —    7    . 


RELATION 
Page  127. 

4-y? 


5.  A 


bu.; 


fbu. 


6.i 


OF    NUMBERS. 


»7       10 

7"  T3 


8.  Given 

9.  24  cts. 

10.  $1093^ 

11.  $264 


218 


ANSWERS. 


Ex.           Ans. 

Ex.          Ans. 

Ex.          Ans. 

12.  Given 

18.  io|£  bu. 

5.  180 

I3-T&T 

6.  261 

14-  T&g 

Page  128. 

7.  46of  J 

15- A 

2.  42I 

8.  $40 

l&irff 

3-56J 

9.  $i6of 

17- $5 

4.  86f 

10.  84  yrs. 

DECIMAL 

I'asre  132. 

1.  .12 

2.  .25 

3- -o5 
.49 


FRACTIONS 


,119 

,027 
,009 
,013 
•1345 


10.  .0236 

11.  .0039 

12.  .OOOO7 

13.  .06  ;  .041 ;  .007 

14.  .0201 ;  .00752 


REDUCTION     OF    DECIMALS. 


Page  133. 

I3-7WU 

5- -3333  + 

1.  Given 

T/1     1881 

6..5 

3-Tto  =  * 

*5*  20000 

7-  -375 

8.4 

9.  .4166-h 
10.  .9 
"••75 

4*  tuvu—^vu 

-      404—101 

5-  -nnnr— if5Tr 

J6-  tAW 
I7.tJ!S* 

18.  iWft 

7- A 

12.  .3125 

8- A 

P«flre  134. 

13.  .2 

9.  AV 

1.  Given 

14.  .0625 

t  t     2  5  2 

2. -5 

15.  .025 
16. . 1 

XI-  Z7ZJ 

3.  .2 

T f       2 

12.  2  oJ0  0 

4.-75 

I  f.    .6 

ADDI1 

ION     OF     DEC! 

MALS. 

IVrflre  J36. 

4.  274.251 

8.  $74-375 

1.  Given 

5.6.6516 

9.  72.946  A. 

2.  33.079 

6.  31.465 

10.  109.841  g. 

3.  16.027 

7.  45.66  yd. 

11.  176.15  r. 

ANSWERS. 


219 


SUBTRACTION     OF     DECIMALS. 


Ex.            Ans. 

Ex.          Ans. 

Ex.           Ans. 

Page  137* 

7.  3.782 

13.  O.045 

2.  3.262 

8.  99.162 

14.  O.O054 

3.  6.1682 

9.  214.25 

15.  O.OOOO9 

4-  27.3797 

10.  7.3992 

16.  6.25  yds. 

5.  O.76442 

11.  14.993 

17.  0.45  ship 

6.  49.525 

12.  0.6306 

18.  62.3  A. 

MULTIPLICATION     OF    DECIMALS. 


Page  139. 

1.  Given 

2.  Given 

3.  Given 

4.  Given 

5.  0.00381 

6.  0.0363 

7.  0.001058 

8.  760.2128 

9.  25.664 


10.  0.3159 

11.  117.351 

12.  18.25 

13.  0.114015 

14.  8.09792 

15.  50.06223 

16.  0.0060024 

17.  53-7758 

18.  70 

19-3 


20.  0.000804 

21.  111.375 

22.  24.375 

23-  393-75 

Page  140. 

24.  10.125 

25.  0.00804 

26.  0.00007 

27.  0.00035 


DIVISION     OF    DECIMALS 


Page  142. 

1.  Given 
2  Given 

3.  Given 

4.  Given 

5.  11  lbs. 

6.  51  lots 

7-  4-312 

8.  0.0^312 

9.  0.0002806 

10.  0.0734201 

11.  142.5 


12.  76 

13.  2.0454  + 

14.  0.4885  + 

15.  0.015 

16.  3.65 

17.  0.0385 

18.  0.5 

19.  0.39104 

20.  2.9029  + 

21.  1000 

22.  100 

23.  0.01 


24.  O.OOOI 

25-  o-75 

26.  $2.5 

27.  4.8  d. 

28.  $0.5 

29.  $10.5 

30.  561.7  +  r. 

31.  181.05 -1  A. 

32.  74  times 

33.  T.066+  t. 

34.  4.2857 +  t. 


220 


ANSWERS, 


ADDITION    OF    U.    S.    MONEY. 


Ex. 


Page  148. 

i,  2.  Given 

3.  $1026.692 

4.  $1631.03 

5.  $2274.52 

6.  $284.37 


Ex. 


7.  $769.73 

8.  $21.00 

9.  $284,375 

10.  $557.43 

11.  $165,846 

12.  $265,525 


Ex. 


Ans. 


13.  $1390.758 

14.  $1967.06 

15.  $3071.58 

16.  $156.13 

17.  $73.50 


SUBTRACTION     OF    U.    S.    MONEY 


Page  149. 

2.  $19,585 

3.  $15,085 

4.  $48,918 

5.  $99,125 


rage  150. 

6.  $0,875 

7.  $20,625 

8.  $1.25 

9.  $187,375 


10.  $83.58 

11.  $990.00 

12.  $160,065 

13.  $94.86 

14.  $296,967 

15.  $19,705 


MULTIPLICATION     OF    U.    S.    MONEY. 


Page  151. 

3-  '$432.85 
$9.01125 
$650,052 
$61987.50 

$22.50 


DIVISI 
Page  153. 

7.  1 1.308 +  t. 

8.  12  times 

9.  63  melons 

10.  85.5  lb. 

11.  $2,125 

12.  1 2 -J  cts. 

13.  14  y  earl's 

14.  80  A. 

15.  #5-375 

16.  $35.63 

17.  $64.03 


8.  $494.00 

9.  $28.1875 

10.  $753-75 

11.  $13500 

12.  $1038 

13.  $31262.50 

ON    OF    U.    S. 

18.  $2.35 

19.  25  cts. 

20.  $9,625 

Page  154. 

1.  $61,895 

2.  $1.75  pro. 

3.  $9.i25dif. 

4.  $38.31  dif. 

5.  $25.84 

6.  $534.60 

7.  17  cts. 


Page  152. 

14.  $756.00 

15.  $19,875 

16.  $78.80 

17.  $186.20  s.; 
$3.80  dif. 

MONEY. 

8.  $7.64$ 

Page  155. 

9.  $10,788  + 

10.  $46.50 

11.  $67.85 

12.  71  hoi^es 
I3.$i5i3.i.5  s.; 

$377,875  a. 

13.  12.5  tubs 

15.  $6 

16.  50O  \YdlV 


ANSWEKS. 


221 


Ex. 


APPLICATIONS    OF     U.    S.    MONEY. 


Ans. 


Page  156. 

i.  $9.59  ami 

2.  $35.25  +$3.1 9 +  $5.04  + $18 +$2 1  =  $82.48,  amt. 

3.  $19.44  +  $16.50  +  $8.80  +  $7.20 =$5 1. 94,  debits; 
$9.00  +  $8.40  +  $8.25  +$18.70  =$44.35,  credits. 

Page  157. 

4.  $i.56  +  $28.5o  +  $4.o8  +  $4.32  +  $io.5o=:$48.96j  amt. 

5.  $16.70 +  $15 +  $9 +  $10.52  =$51.22,  debits; 
$24.00  +  ^36  +  $62  +$26.25  —$148.25,  credits ; 
Balance  due  J.  Barker,       =$97.03 

6.  $47.34  + $24.50 +  $28.50 +  $9.75 +$n.io  +  $7.5o  + 
$10.80 =$139.44  amt. 


Page  174. 

3.  8714  far. 

4.  835  far. 

5.  io368d. 

6.  1 1 0400  far. 

Page  175. 

9.  £6,  is.  9d. 

10.  £28,  3s.  2d.  3  f. 

11.  Given 

1 2.  £3,  8s.  9d. 

13.  1780  pwt. 

14.  61 1 13  gr. 

15.  61b.  6  oz.  1  pwt, 

16.  1  lb.  1  oz.  13  p. 
23  gr. 

17.  1  oz.  18  pwt. 

18.  $78 

19.  1  cwt.  41  lb. 
9  oz. 

20.  483704  oz. 


REDUCTION. 

21.  3201204  OZ. 

22.  $5.04 

23.  $656.25 

24.  480  dr. 

25.  21 12  sc. 

26.  1  lb.  7  oz.  4  dr. 
1  sc. 

27.  30Z.  2  dr.  18  g. 

28.  742  J  ft 

29-  63532  ft- 

30.  46422  yd. 

Page  176. 

31.  Given 

32.  94  r.  9  ft. 

33.  2  m.  43  r.  8J  ft. 

34.  1 7 1049  ft. 

35.  $2.25 

36.  1280  rods 

37.  10560  st. 

38.  5  quarters 


39.  6  eighths 

40.  2416  i6ths 

41.  118J  yd. 

42.  $3.20 

43.  $420 

44.  43560  sq.  ft. 

45.  234407 J  sq.ft. 

46.  102729  sq.  yd. 

47.  102400  sq.  r. 

48.  5  A.  51  sq.  r. 

49.  15  A.  100  sq.  r. 

50.  35  s.y.  6  sq.ft. 
49  sq.  in. 

Page  177. 

51.  $1361.25 

52.  $10454.40 

53.  1940544  cu.  in. 

54.  2  cu.  y.  1  cu.  ft 
1325  cu.  in. 

55.  33  C.  26  cu.  ft 


222 


ANSWERS. 


Ex. 


Ans. 


56.  9664  cu.  ft. 

57.  1943  cu.  ft. 

58.  $486 

59.  $76.80 

60.  $84 

61.  442  pts. 

62.  51  bu.  1  p.  7 

63.  $19.20 

64.  $3.90  prof. 

65.  136  boxes 

66.  3  bu.  1  p.  3  ( 

67.  207  gills 

68.  770  qt. 

69.  57  gaL  1  q. 


Ex. 


Ans. 


70.  10  hhd.  10  g. 

71.  45  qt. 

72.  $63 

Page  178. 

73.  $142.80 

74.  278220  sec. 

75.  3d.4h.5m. 

76.  77760  m. 

77.  525600  m. 

78.  5y.185d.16h. 

79.  604800  t. 

80.  $294 

81.  3258720  t. 


Ex. 


Ans. 


82.  11  d.  S^/yh. 

83.  16*780" 

84.  20  46'  40" 

85.4  s.  17°  55' 

86.  15"  in  1  h.; 
i°  in  4  m. 

87.  2640  sh. 

88.  25  r.  10  qu. 
1 8  sh. 

89.  $0,005^ 

90.  5760  cra)rons 

91.  $126 

92.  864  pens 
93-  75  eggs 


SURFACES    AND     SOLIDS. 
Page  179. 


1.  Given 

2.  28  sq.  ft. 

3.  768  sq.  in. 

4.  90  sq.  r. 
5-  3°  yds. 

Page  180. 

6.  24  sq.  ft. 

7.  50  A. 


8.  23040  A. 

9.  540  sq.  yd. 

10.  900  brick 

11.  $3.60 

Page  181. 

1.  Given 

2.  64  cu.  in. 

3.  45  cu.  ft. 

4.  240  blocks 


5.  455  cu.  ft 

6.  98  cu.  ft. 

7.  378  cu.  ft. 

8.  160  cu.  yd. 

9.  81  cu.  ft. 

10.  $243 

11.  $40.50 

12.  $40.50 

13.  240  cu.  ft. 

14.  90  cu.  ft. 


REDUCTION 

Page  182. 

1.  Given 

2.  1  ft.  io|  in. 

3.  78.  6d. 

4.  5  d.  6  hr. 

Page  183. 

6.  §  qt. 


OF    DENOMINATE    FRACTIONS 


15.  £f 

16.  Given 

18.  .^bu. 

19.  Given 

20.  0.38125  m. 

21.  £0.33$ 

22.  0.09-^  hhd. 


7.  f  pwt. 

9.  3  qt.  .72  pt. 

10.  4  d.  9  hr. 

11.  3  pk.  4qt. 

Page  184. 

13-  tt  yd. 

14.  &  bu. 


ANSWERS. 
COMPOUND    ADDITION 


Ex.        Ans. 

Ex.        Ans. 

Page  186. 

7.  186  m.  24  r. 

2.  £26,  28.  id.  3  far. 

8.  92  bu.  1  pk.  4  qt. 

3.  27  lb.  11  oz. 

9.  H7iyd. 

4.  22  yd.  8  in. 

10.  54  sq.  r.  19  sq.  yd.  7  sq.  ft 

5.  108  bu.  3  pk.  6  qt. 

6.  25  T.  5  cwt.  23  lb. 

1  pt. 

11.  117  A.  48  sq.  r. 

1 2.  4  C.  60  cu.  ft. 

COMPOUND     SUBTRACTION. 


Page  187. 

1.  Griven 

2.  £2,  2s.  2d.  3  far. 

3.  5  lb.  11  oz.  5  pwt.5  gr. 

4.  1  bu.  o  p.  3  qt. 

5.  7  m.  199  r.  i}y.  1  ft, 
or  7  m.  199  r.  1  y.  2  ft. 
6  in. 

6.  27  gal.  1  qt. 


Page  188. 

7.  64  A.  143  sq.  r. 

8.  41  cu.  ft. 

9.  2  T.  252  lb. 

10.  13°  34' 57" 

11.  15°  33'  30" 

13.  26  yr.  6  mo.  3  d. 

14.  1  yr.  11  mo.  12  d. 

15.  4  yr.  2  mo.  24  d. 

16.  3  y.  10  m.  23  d. 


COMPOUND    MULTIPLICATION. 


Page  190. 

4.  184  g.  1  q.  1  p. 

5.  256  yd.  2  qr. 


6.  61  A.  92  sq.  r. 

7.  12  C.  94  c.  ft. 

8.  67  hr. 


9.  9  T.  625  lb. 

10.  £70,  4s.  4d. 

11.  85  bu.  1  pk. 


COMPOUND     DIVISION. 

rage  192. 

3.  3  A.  47  sq.  r. 
7  sq.  ft. 

4.  84  cu.  ft.  31 3§ 
cu.  in. 

5-  7i  I2  g* 


6.9 

7.  6  bu.  2\  pk. 

8.  10  ft.  4  in. 

9.  5  m.  3  fur. 
10.  2  oz.  12  pwt. 


11.  4400  rails 

12.  6  bu.  1 J  pk. 

13.  2  A.  24  sq.  r 


..  24  sq.  r. 

14.  640  times 

15.  7  bags 

16.  2  iff  bundles 


224 


ANSWERS, 


PERCENTAGE. 

Ex.           Ans. 

Ex.           Ans. 

Ex.          Ans. 

Page  1 96. 

13.  600  pupils. 

II.    25%. 

i,  3.  Given. 

14.    $15625. 

12.    25%. 

4.  32  yds. 

13.    25%. 

5.  $45-936- 

rage  198. 

14.    25%. 

6.  42  bu. 

1,  2.  Given. 

15.    50%. 

7.  80  rods. 

3-  33i#. 

16.  33i?o- 

8.  133.2  bar. 

4-  3°°^- 

9.  408  men. 

5.  20%. 

Page  200, 

6.   1 1  \%. 

1,  2.  Given. 

Page  197. 

7.   50%. 

3.  $62.50. 

10, 11.  Given. 

8.  33i%- 

4.  $47,024. 

12.  81680. 

9.  20$. 

5,  6.  Given. 

10.   i6f#.     • 

7-  $35- 

PROFIT    AN  D     LOSS. 

Page  202. 

9,  10.  Given. 

2.  25%. 

1,  2.  Given. 

11.  $93.60. 

3-  5°%- 

3.  $7  gain. 

12.  $336. 

4.  50 % 

4.  $0.75. 

13.  $2025. 

5.  28}%. 

5.  $2,125. 

14.  $5600. 

6.   isifo 

6.  $105.30. 

7-  33i7°- 

7.  $18.40. 

Page  203. 

8.  25%. 

8.  $65. 

1.  Given. 
INTEREST. 

9.    427#- 

JPaflN?  207. 

21.  $80.00  int. ; 

28.  Given. 

1- 14.  Given. 

$580    amt. 

29.  5  m.  9  d. time; 

15.  $5,176. 

22.  $69,646   int. ; 

$33,125  int.; 

16.  $6,668    int.; 

$832,897  amt. 

$1283. 1 25  am. 

$65,418  amt 

23.  $360. 

30.   10  m.  24  d.  t.; 

17.  $2.94. 

24.  $306,683. 

$126.00  int.; 

18.  $4,294. 

25.  $15,125. 

$2126    amt. 

19.  $11.25. 

26.  $45-325- 

31.  $1350  int.; 

20.  £25.175. 

27.  $32,301. 

$5850  amt. 

YB  35847 


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